But there are also growing rumors that Scott might be in the running for a higher-profile position than taking one of 100 seats in the U.S. Senate — that he could instead be the running mate for Donald Trump if the real-estate mogul clinches the GOP presidential nomination, as seems increasingly likely.
Trump himself encouraged the rumors when, in an interview with The New York Times for a story published this weekend about a possible vice presidential selection, "he briefly praised three governors as possible contenders," including Scott. The other two names Trump listed were New Jersey Gov. Chris Christie, one of the first major Republicans to endorse Trump, and Ohio Gov. John Kasich, who's still running for president.
Other media outlets have also contributed to the speculation. Writing last week on the website of the conservative National Review, Jim Geraghty said Scott "might be the most intriguing choice" for Trump's running mate.
"Scott won two exceptionally nasty, hard-fought gubernatorial races and could lock up the state for Trump — giving him a win that John McCain and Mitt Romney could not achieve," Geraghty wrote. "Scott's a relatively low-key, even-tempered figure among Trump endorsers; he may not have an overwhelming personality, but he isn't likely to do much harm."
The governor has kept up a rigorous travel schedule that would seem aimed at raising his national profile. Scott is currently in California, one of several job-poaching trips he's taken to states with Democratic governors. Over the weekend, he attended the White House Correspondents Dinner, one of the biggest annual social soirees in the nation's capital.
"Scott's a relatively low-key, even-tempered figure among Trump endorsers; he may not have an overwhelming personality, but he isn't likely to do much harm."
But Melissa Sellers, Scott's former chief of staff and a current adviser, dismissed the speculation Monday.
"The governor has made it clear that he has the job he wants and he won't leave it," Sellers said in an email. "Of course, he would always be willing to provide any insights from his experience in turning around Florida's economy. Governor Scott is a long-time friend of Donald Trump's. They have known each other in the business world for many years — long before the governor first ran for office in 2010."
In some ways, an alliance between the two men would make sense. Like Trump, Scott is an outsider businessman who fought the party establishment in his first bid for public office.
"The common thread, of course, is the economy and 'jobs, jobs, jobs,' " said Susan MacManus, a political science professor at the University of South Florida.
And since Trump won the Florida presidential primary on March 15, Scott has become one of his stronger supporters — calling last week for the "Stop Trump" movement to give up after Trump swept primaries in mid-Atlantic states.
"Donald Trump is going to be our nominee, and he is going to be on the ballot as the Republican candidate for President. ... If the anti-Trump groups don't stop now, their efforts will be nothing more than a contribution to the Clinton campaign," Scott wrote on his Facebook page.
There are questions, though, about how much either man might be able to help the other. Scott has long been a divisive governor in Florida. In the most recent Quinnipiac University Poll to survey Floridians' attitudes about Scott — taken in October — 41 percent of voters approved of the governor's performance, compared to 47 percent who disapproved.
Scott Arceneaux, executive director of the Florida Democratic Party, had a simple response to the prospect of a Trump-Scott ticket, according to a spokesman: "We could not be so lucky."
And while Scott narrowly won two terms as governor, he did so in midterm elections. Democratic voters tend to turn out in higher numbers in presidential elections.
At the same time, if Scott were to tie himself too closely to Trump only to see the state won by Hillary Clinton, the likely Democratic nominee, he could damage a potential Senate campaign in two years.
"I'm not sure about the pragmatism for either of them," MacManus said.
MacManus also said that naming Scott as a possible running mate might also simply be a nod toward a critically important swing state in the fall. She noted that Nelson has also been brought up as a contender for Clinton's No. 2 spot.
"Mentioning Florida is good politics," she said.