HOUSTON — A comment President Trump made on Tuesday in which he spoke of destroying the career of a Texas state senator has angered Democratic lawmakers in the state and cast a spotlight on the issue that set off the remark — the seizing of suspects’ property by law enforcement before those suspects have been convicted of a crime.

Mr. Trump’s remark came during a meeting at the White House with sheriffs from around the country. At one point, the president asked the sheriffs seated at a table around him if there were any pressing law enforcement issues they wanted to talk about.

A Texas sheriff, Harold Eavenson of Rockwall County, spoke up.

“Mr. President, on asset forfeiture,” Sheriff Eavenson said, in an exchange that was observed by reporters and filmed, “we’ve got a state senator in Texas that was talking about introducing legislation to require conviction before we can receive that forfeiture money, and I told him that the cartel would build a monument to him in Mexico if he could get that legislation passed.”

“Can you believe that?” Mr. Trump responded, then added, “Who’s the state senator?”

Sheriff Eavenson did not reply. “Do you want to give his name?” Mr. Trump said. “We’ll destroy his career.” Laughter then broke out.

It remained unclear which Texas state senator Sheriff Eavenson was talking about. The sheriff, one of the top officials in a county about 25 miles east of Dallas, did not respond to requests for comment on Wednesday.

The state senator who represents Rockwall County, Bob Hall, a Republican, also did not respond to requests for comment. According to The Texas Tribune, Mr. Hall told reporters on Tuesday at the state Capitol in Austin that he had “no idea” if he was the senator the sheriff was referring to, but said he had a “very good relationship” with the sheriff.

Mr. Hall is a member of the Senate committee on veteran affairs and border security, and is a decorated Air Force veteran who worked as an engineer on the Minuteman missile-defense system during the Cold War.

Texas state senators and representatives reacted to Mr. Trump’s remark along party lines. Republicans, who control both chambers of the State Legislature, described Mr. Trump’s comment as a joke. Democrats, however, said they were shocked that the president of the United States would speak so flippantly about destroying a lawmaker’s career.

State Senator José Menéndez, a Democrat from San Antonio, wrote a proposed resolution condemning Mr. Trump’s comment. The resolution, which was filed but had not been voted on, states that when the president “threatens any member of the Texas Senate, it must be considered a threat to all Texas senators,” and it calls on Mr. Trump “to refrain from threatening elected officials.”

Both of the senators who filed bills in the Texas Legislature to restrict how and when property can be seized by law enforcement denied they were the ones whom the sheriff had been referring to. One of them, Konni Burton, a Republican, said she disagreed with the sheriff’s comments about overhauling asset forfeiture, and said the incident would not discourage her from pursuing passage of her bill, which requires conviction before assets can be seized.

“Do not be mistaken or misled: This is not strictly a law enforcement issue,” Ms. Burton said in a statement. “This is a property rights issue. Property rights are one of the foundational rights in any free society, and the taking of property by government is no small matter.’’

“Requiring the government to secure a criminal conviction before permanently taking property from citizens is simply common sense,” she said.

The other senator who filed legislation on the issue, Juan Hinojosa, a Democrat, said the “bills are an important protection for Texans’ property rights and civil liberties.” He added, “Quite frankly, I don’t pay much attention to what President Trump says anymore.”

The effort to overhaul and tighten the state’s laws that allow law enforcement to seize suspects’ cash or property has won support from Democrats and Republicans. Critics of the practice say it has led to widespread violations of property rights in several states, while law enforcement groups say it remains an effective tool to hurt drug cartels.

Ms. Burton said that in Texas, “many cases of asset forfeiture end without the state securing a criminal conviction, but the state still successfully keeps most, if not all, of the property.”

In a statement Sheriff Eavenson posted on Facebook, he said that in bringing up the unnamed state senator’s position, he sought “to make the point public to possibly benefit law enforcement.”

“My personal opinion is that such a bill, if it were to pass, would benefit the cartels and damage law enforcement,” he said.

Texas has long taken threats against state senators and other elected officials seriously.

One such threat made by Rick Perry, then the state’s governor, when he threatened to veto state money for the public corruption unit of the Travis County district attorney unless she resigned after a drunken-driving arrest, was at the center of an indictment against him in 2014. The case was dismissed two years later.

Michael McCrum, a San Antonio lawyer and former federal prosecutor who served as the special prosecutor in the Perry case, said he believed Mr. Trump’s comment came close to crossing the line from a joke to a criminal act.

“Did he knowingly threaten to harm another?” Mr. McCrum said. “It’s certainly up close to the line. I don’t think it’s over the line.”

Susan R. Klein, a professor at the University of Texas at Austin School of Law and a former federal prosecutor who helped prepare Mr. Perry’s legal team for oral arguments, said Mr. Trump’s remark was clearly a joke and did not violate state or federal law.

“I am a liberal Democrat who voted for Hillary Clinton, but I really think we should let up on Trump now and accept he is our president and wish him well,” Professor Klein said in a statement. “Trying to obstruct him at every turn and even criminally charge him for what was clearly a joke is not helping this country one bit.”