Still, even some of Mr. Lee’s backers note the contrast in the post-shutdown reception given to Mr. Cruz, who according to news accounts received an eight-minute ovation from hundreds of people at a Texas GOP women’s group meeting Saturday, and Mr. Lee.

“Republicans here are polarized, no question about it,” said Spencer Stokes, who served for two years as Mr. Lee’s Senate chief of staff. “So, Ted Cruz went home to a standing ovation, and Mike Lee has hardly wanted to go shopping for fear of being confronted.”

Republican circles are now rife with talk of who might challenge Mr. Lee in 2016. So far, no one is firmly raising a hand. But the Count My Vote initiative to do away with the state’s caucus system, backed by many of the state’s largest GOP donors and business names, represents perhaps the best-organized effort in the country to counteract the tea-party wave in the 2010 elections.

Mr. Lee could face a tougher route to re-election in 2016 if GOP caucuses are replaced with a direct primary. That would allow a more centrist candidate to make an appeal to all Republican voters, not just the activists who dominate caucuses, political observers say…

“Lee looks vulnerable to a challenge from within his party, but the real danger could be a challenge in a general election from the right kind of moderate Democrat,” said Quin Monson, who directs the Utah Voter Poll at Brigham Young University.