After the first votes were counted in Tuesday’s primary, Mr Kent, 42, was some way behind the incumbent, as well as Democratic candidate Marie Perez in the race for Washington’s third congressional district.
But as the count has continued, with more results being released, Mr Kent has seen his numbers soar.
On Friday evening, with around 80 per cent of all votes counted and as officials announced there would be no more results released until Monday, Mr Kent was just 257 behind Ms Herrera Beutler.
“There will be no counting over the next two days, because we believe in weekends here in Washington state (thank a union!),” tweeted Phil Gardner, the chief of staff to Washington’s Lt Governor.
“Next reports will come from Clark, Cowlitz, and Thurston on Monday evening.”
Ms Perez, 34, the Democrat, saw her total increase to 57,345, putting her on 31.2 points, ahead of Ms Herrera Beutler, 43, on 22.6 and Mr Kent on 22.5.
The Associated Press had already called the race for Ms Perez, but it was unclear who she would face off against in the general election in November.
Washington state is one of just a handful of places that make use of open top-two primaries in which the highest scoring candidates – regardless of party – go on to the general election.
Ms Herrera Beutler was first elected to Washington’s third congressional district in 2010 and reelected five times.
But she was one of just ten Republicans in the House who voted for Mr Trump’s impeachment after the Jan 6 riots and the establishment of a committee to investigate what happened.
Mr Trump denounced the ten as traitors and vowed to oust them all.
Another of the ten, Dan Newhouse, from Washington’s fourth congressional district, held off a challenge from Loren Culp, a former police chief who had the ex-president’s backing in that contest. He will go to the general election against Democrat Doug White, the Associated Press announced on Friday.
On Tuesday night, Michigan congressman Peter Meijer lost to challenger John Gibbs, who had the support of Mr Trump.
Two others had primaries in June. Tom Rice of South Colorado, while David Valadao in California survived. Mr Trump’s main nemesis, Liz Cheney, faces a primary showdown on Aug 16. She is widely expected to be defeated.
Political analyst Dave Wasserman of the Cook Political Report tweeted: “Pro-impeachment Rep Jaime Herrera Beutler needs a miracle. Trump-endorsed Joe Kent is now the strong favorite to knock her out of the top two.”