Mitt Romney's son, Josh, could be competitive if he decided to run for Utah Sen. Mike Lee's (R) Senate seat in 2016, according to a new poll.

Among Republican Utah voters, 49 percent said they would vote for Lee if the Republican primary were today, while 36 percent said they would vote for Romney, the Utah Policy poll found. An additional 14 percent said they were undecided.

Romney has said he wouldn't rule out running for office, and Lee has seen his approval ratings fluctuate since he took office in 2011. During the government shutdown in 2013, when Lee sided with Sen. Ted Cruz (R-Tex.) in favor of holding out to try and defund Obamacare, he had a 40 percent approval rating. By September 2014, it rose to 54 percent. At the time, University of Utah professor Matthew Burbank told the Salt Lake Tribune that Lee's popularity increased because he wasn't in the news as much.

"When he was in the news in the past, it tended to be around issues like the shutdown, which were broadly unpopular," he said. "There is always a dissipation as we get farther away from those events."

Former Utah GOP chairman Thomas Wright received 1 percent in the poll, while former governor Jon Huntsman (R), a big name floated as a possible candidate, was not included. Huntsman has said he would not run for the seat, even though his father called Lee "an embarrassment to the state of Utah."

The poll found the favorite Democrat was former congressman Jim Matheson (D), with 54 percent. Matheson joined the board of student loan lender Sallie Mae in March.

Incumbency is always powerful in elections, but in Utah, so is the Romney name. There aren't very many politicians whose endorsement would carry as much weight in Utah as Mitt's would, especially given his role in the 2002 Salt Lake City Olympics.

Of course, sometimes a Romney endorsement isn't enough. In 2009, Mitt endorsed Lee's predecessor, then-Sen. Bob Bennett, and Bennett came in third in the state GOP convention, not even qualifying for the primary. Conventions can be much more conservative than a typical party primary, which could give Lee an edge over opponents, but they are also highly unpredictable.

But right now, it's not entirely clear just how the nomination process will work. A change in the law that allowed candidates to bypass the convention and collect signatures to get on the primary ballot is being challenged by the state GOP, and a judge is set to rule soon.