Representative Mike Conaway will not seek re-election to his Texas seat in 2020, the Politico website reported. He has not confirmed his decision or reason for retiring but he is set to make a statement to the media.
The move has prompted worries within the party that others will follow suit and step down, because of the difficulties that come with serving under Mr Trump and working with a Democratic majority in Congress.
“Serving in the era of Trump has few rewards,” Tom Davis, a former senior Republican congressman, told The Hill website. “He has made an already hostile political environment worse.
“Every day there is some indefensible tweet or comment to defend or explain. It is exhausting and often embarrassing.”
Mr Conaway, who has served in congress for 15 years, will join Republican representatives Paul Mitchell, Pete Olson, Martha Roby and Rob Bishop in announcing his retirement.
Mr Mitchell had told the House that “rhetoric overwhelms policy and politics consumes much of the oxygen” in Washington DC.
One of his former campaign workers Jamie Roe, later said that Mitchell had “been frustrated with the fact that things don’t get done here”.
While he did not explicitly attribute blame to the president, he was one of the first Republican congressmen to complain about Mr Trump’s recent racist remarks about the group of Democratic congresswoman known as the squad.
“We must be better than comments like these,” he tweeted after the president suggested Alexandria Ocasio-Cortez of New York, Ilhan Omar of Minnesota, Ayanna Pressley of Massachusetts and Rashida Tlaib of Michigan should go back to their “broken and crime-infested” countries.
Mr Mitchell added: “I share the political frustrations with some members of the other party, but these comments are beneath leaders.”
Ms Roby said she would not vote for Mr Trump in 2016 as his behaviour had been “unacceptable as a candidate for president” but has since improved their relationship and received an endorsement from him in 2018.
The Republican Party is facing a difficult task in reclaiming the House in 2020 after Democrats were victorious in last year’s midterm elections.
Mr Trump’s approval ratings remain low, currently at about 43 per cent on average, and his divisive political agenda could prove costly in congressional elections next year.
Mr Conaway, Mr Mitchell, Ms Roby and Mr Bishop all represent safe Republican districts that are expected to pick candidates from the party in 2020.
However, Mr Olson’s district could be competitive – the Texas congressman saw his majority cut to 5 per cent in 2018.
Even in safe districts, the prospect of returning to the House in 2020 may be unappealing for many conservative representatives as Democrats are expected to win a majority again next year.
In a general ballot, recent polling has shown Democrats lead Republicans by 5.6 per cent for the 2020 election, according to an average by political analysis website FiveThirtyEight.
All 435 voting seats in the House of Representatives will be up for election in 2020, along with 34 seats in the US Senate.