Qatar Airways Ltd. is seeking to buy about 10 percent of Fort Worth-based American Airlines Group Inc. — an unsolicited overture by a bitter rival that would give the Middle Eastern carrier a significant stake in the world’s largest airline.

The move by state-owned Qatar Airways comes as American (Nasdaq: AAL) and other major U.S. carriers accuse Qatar and other Gulf airlines of violating air treaties, allegedly giving them an unfair advantage. Specifically, the U.S. carriers claim the Gulf…