Representative Dan Goldman, Democrat of New York, will introduce a resolution to affirm Congress’s support for the 22nd Amendment limiting presidents to two terms in office, as President Trump and his allies continue to talk about pushing for a third term.

“Once dismissed as a ‘joke’ by the media, Donald Trump’s repeated allusions to serving beyond his constitutionally mandated second term have now become standard G.O.P. orthodoxy,” Mr. Goldman said Thursday.

Mr. Goldman said he is exploring ways to force a vote on the resolution, even though Democrats are in the minority in the House. This could include using a tactic known as a discharge petition, which can force consideration of a bill if it has enough backing from members of Congress and would put Republicans on the record.

Few Republicans have been willing to break publicly with Mr. Trump, though some have described his musings about a third term as merely a joke.

The G.O.P. does not have enough votes to amend the Constitution, which would require a two-thirds vote of Congress and three-fourths vote of the states. But that hasn’t stopped some on the right and the president himself from musing about a third term.

Representative Andy Ogles, Republican of Tennessee, has already introduced legislation to amend the Constitution to allow Mr. Trump to run for a third term. A group calling itself the Third Term Project is attempting to rally support for the legislation. During the Conservative Political Action Conference this month, signs depicted Mr. Trump as a Roman emperor with the caption: “For Trump 2028 … And Beyond!”

In recent weeks, Mr. Trump has repeatedly spoken about another term, and he has referred to himself as a “king” on his social media site. Mr. Trump’s advisers say he is merely trolling his critics, but in office he has taken a maximalist view of his executive authority.

The president is also challenging another constitutionally protected right, birthright citizenship. But several federal judges have issued injunctions against that executive order, which would end automatic citizenship for the children of undocumented immigrants born in the United States.