Establishment figures in Utah have long loathed the convention system and are launching a well-funded effort to change it. A bipartisan group including former governor Michael Leavitt, a Republican who was a George W. Bush Cabinet official and close Romney adviser, has launched Count My Vote, a ballot initiative to overhaul Utah's nominating process. The group has raised more than $500,000, most from major GOP donors.
A shift to an open primary could hurt Lee, who supports the convention system because his most passionate supporters are the conservative activists who become delegates. Rich McKeown, a longtime Leavitt aide and chairman of the effort, insisted that Count My Vote is designed not to target the senator, but rather to enlarge the voting population over the long term.
One beneficiary could be Thomas Wright, who stepped down this spring as chairman of the Utah Republican Party. Wright said he is considering running against Lee in 2016 because he has grown "exasperated" with the junior senator's governing style.
"We can't keep going on like this," Wright said. "I want to work with people to get things done. I want to go be a leader and build bridges, not burn them down."
Former state senator Dan Liljenquist and Josh Romney, one of Mitt's sons, have also been mentioned as possible challengers, Utah Republicans say. Liljenquist enjoyed tea party backing when he ran unsuccessfully against Sen. Orrin Hatch in last year's Republican primary.
Liljenquist criticized Lee's handling of the shutdown. "I'm struggling to see what was gained from it for Utah," he said. "An all-or-nothing approach makes people uncomfortable here."
Although Cruz attracts more attention, Lee is one of the main intellectual forces behind the tea party in Washington. His Utah supporters say they're proud that he is uncompromising.