Gov. Matt Bevin of Kentucky was unrestrained in his praise for President Trump: Opening for him at a rally on Monday, Mr. Bevin, a conservative Republican, echoed Mr. Trump’s “America First” slogan and only gently noted the nagging divisions in their party.

“We now have a president and a Congress that are united in party, and yet we still have disagreements among us,” Mr. Bevin said, insisting, “This is healthy and good.”

In private, Mr. Bevin has been blunter about the party’s disagreements. Just days before appearing with Mr. Trump in Louisville, he joined a conference call with the president’s budget director, Mick Mulvaney, to protest a White House proposal to defund the Appalachian Regional Commission, an economic development agency that spans 13 states and steers millions of dollars in federal money to Kentucky.

Mr. Bevin was not alone in his dismay.

As Mr. Trump and his advisers press for bone-deep cuts to the federal budget, Republican governors have rapidly emerged as an influential bloc of opposition. They have complained to the White House about reductions they see as harmful or arbitrary, and they plan to pressure members of Congress from their states to oppose them.

Of acute concern to Republicans are a handful of low-profile programs aimed at job training and economic revitalization, including regional development agencies like the Appalachian commission and the Delta Regional Authority, which serves eight Southern and Midwestern states, seven of them with Republican governors. They are also protective of grants from the Department of Housing and Urban Development and a $3.4 billion job-training program funded through the Labor Department.

Mr. Trump’s budget office has proposed to eliminate or deeply slash funding for all of those programs, along with dozens of others.

Kim S. Rueben, a budget expert at the nonpartisan Tax Policy Center, said the retrenchment in Mr. Trump’s spending plan appeared to be significantly out of step with his campaign promises to use the federal government as a machine for creating jobs, especially in distressed Midwestern and rural areas.

“It just seems like you’re going after places that are so pivotal to what you are arguing you wanted to do for your base,” Ms. Rueben said of Mr. Trump’s budget. “They’re cutting all sorts of infrastructure projects and economic development projects at the same time that the president is still talking about how much of an investment he’s going to put into infrastructure.”

The White House’s proposed cuts would be felt in matters well beyond economic development: A budget briefing circulated last week by the National Governors Association, a nonpartisan group, identified a long list of Trump-backed cuts to programs that support states. They include the Low Income Home Energy Assistance Program, a $3 billion project in the Department of Health and Human Services that helps people pay for heating and air conditioning, and the Community Development Block Grant program, a $3 billion initiative of the Department of Housing and Urban Development that funds local projects from affordable housing to Meals on Wheels.

Document | National Governors Association Warns of Proposed Federal Cuts A budget briefing circulated last week by the National Governors Association, a nonpartisan group, identified a long list of Trump-backed cuts to programs that support states.

Those cuts could come on top of a potentially huge restructuring of the federal Medicaid program under a Republican-backed health care law. A number of Republican governors, including John Kasich of Ohio and Brian Sandoval of Nevada, have publicly criticized the bill under consideration in the House of Representatives because they say it would impose an impossible fiscal burden on states.

Republicans have long argued for a more limited federal role in matters of economic engineering and social welfare, preferring to collect less tax revenue at the national level and hand over responsibility for a range of programs to state and local governments. But in practice, state leaders in both parties often balk at taking on such burdens.

Some of the governors who have voiced worry about the White House budget are among the country’s most conservative. Far from welcoming additional responsibilities, many of them have focused intently on limiting the size and cost of state government.

Gov. Robert Bentley of Alabama said he intended to push back against planned cuts to the Appalachian and Mississippi Delta economic agencies, as well as to the community development grants.

“The Appalachian Regional Commission, the Delta Regional Authority and the Community Development Block Grants are important resources that provide funding that benefits rural projects such as infrastructure improvement, job creation, technology upgrades and school programs,” Mr. Bentley said in a statement. “Along with my governor colleagues in the A.R.C. and D.R.A., I look forward to sharing with Washington how vital these assets are to our poorest and smallest communities.”

Gov. Asa Hutchinson of Arkansas has already told the administration his objections to its plans for the Delta Regional Authority, which gave $10 million in federal grants to states last year, and which the White House budget would eliminate.

Mr. Hutchinson “wants to make sure the Delta is not cut off from necessary economic development funding,” said J. R. Davis, a spokesman for the governor. “It’s a relied upon program.”

So far, the administration has no apparent strategy to placate uneasy Republican governors. When Mr. Mulvaney, the budget director, briefed a bipartisan group of governors and their aides by telephone last week, he spent just a few minutes on a broad synopsis of the administration’s spending plans and took no questions before turning the call over to a member of the White House staff, according to four people on the call, some of whom spoke on the condition of anonymity because it was intended to be private.

By way of defending such extensive cuts, Mr. Mulvaney said simply that the White House’s priority was military spending and that other reductions were necessary to advance that goal.

After Mr. Mulvaney left the call, White House aides spent about 20 minutes taking down polite complaints from leaders in both parties, including Republicans like Mr. Bevin and Paul R. LePage, the governor of Maine and a strong supporter of Mr. Trump, who said he was concerned about cuts to housing for the poor. (Spokeswomen for Mr. Bevin and Mr. LePage declined to comment.)

But the aides offered no clear reassurances to the governors, according to people on the call, merely pledging over and over to get back with answers to their questions.

“It was, ‘We’ll get back to you on that,’” recalled Pat Pitney, the budget director for Gov. Bill Walker of Alaska, a former Republican who was elected as an independent.

Ms. Pitney said that there was considerable skepticism among governors that Mr. Trump would be able to enact his budget, but that states heavily dependent on federal money were already forming coalitions to oppose some provisions. Many of those alliances are likely to transcend partisan divisions, Ms. Pitney said, as states team up on matters of regional importance.

“There are going to be a lot of these things that don’t fall on party lines, because it’s so impactful to the communities that the Congress and the Senate represent,” Ms. Pitney predicted, adding of the proposed cuts, “I think there are some that are dead on arrival, but not every one is dead on arrival.”

For all their private unease, only a few Republican governors have openly criticized Mr. Trump’s budget. Advisers to several of them, including some who spoke on the condition of anonymity, said they saw little benefit to doing so, since they believe the budget is all but doomed in Congress already. Leaders in both the House and Senate have indicated that they are unlikely to pass it in anything resembling its current form, and some have already lined up with their party’s restive governors: Senator Mitch McConnell of Kentucky, the majority leader, has publicly vowed to oppose any cuts to the Appalachian Regional Commission of the kind Mr. Bevin fears.

Aides to multiple governors, in both red and blue states, signaled that they doubted Congress would pass any budget at all, let alone one as disruptive as Mr. Trump’s.

Stephanie Wilson, a spokeswoman for Gov. Eric Holcomb of Indiana, a Republican elected in November, said the governor’s office would work closely with Indiana’s largely Republican congressional delegation to defend state priorities.

“But we understand the budget process is a long one,” Ms. Wilson cautioned in an email. “The president’s proposal will need to work its way through Congress.”

Amelia Chassé, a spokeswoman for Gov. Larry Hogan of Maryland, was perhaps even more direct: Asked about Mr. Trump’s proposal to wipe out funding for cleanup of the Chesapeake Bay, Ms. Chassé said Mr. Hogan, a Republican, supported restoring the bay and would address the White House budget as necessary.

“If any of these proposals ever become law or even draft legislation,” she said, “we will take a serious look at how to address them during our own budget process.”