Former Republican National Committee member Hageman is leading the way with 52 percent of the public support according to the poll, while three-term incumbent Cheney's support is at 30 percent.
The results come less than a month before the Aug. 16 primary for the Wyoming House GOP primary is set to begin.
The survey questioned 1,100 registered Wyoming voters likely to participate in the primary, with a margin of error of 3 percent.
Cheney is facing a tough election in the bid for the House seat following her decision to serve as vice-chair on the Democrat-controlled House subcommittee investigating the Jan. 6, 2021 attack on the U.S.Capitol.
She has also previously been vocal in her criticism of former President Donald Trump, and voted to impeach him in his second trial. That move prompted the Wyoming Republican Party last year to vote to censure her before deciding to stop recognizing her as a member.
Problems at Home"Wyoming politics is very personal, it’s face-to-face, it’s town-to-town," Barrasso told the publication. "The travel that I have done around the state, I think she has a lot of work to do if she hopes to win the primary."
Trump himself has said he believes Cheney will lose her reelection bid, pointing to Rep. Tom Rice (R-S.C.), who lost by 28 percentage points in the GOP primary several weeks ago. Republican Rice also voted to impeach Trump.
Meanwhile, Cheney's campaign last month began turning to the state's Democrats for support, informing them via her campaign website on to switch parties to vote for her in the GOP primary.
However, Cheney's allies believe her outspokenness about Trump may have put her in a stronger position for the 2024 presidential contest; something the lawmaker has not yet ruled out.
The Republican lawmaker added that she will continue to "fight hard" to ensure that Trump is "never anywhere close to the Oval Office ever again."