(CNN) Democrats in Washington and Kentucky, eager to seize on Sen. Rand Paul's stumbles in the Republican presidential race, are increasingly setting their sights on what once seemed unthinkable: Knocking him out of his Senate seat.

The prospects, they acknowledge, are still slim -- given Kentucky's sharp tilt to the right in recent years. But behind the scenes, Democrats are urging the wealthy mayor of Lexington, Jim Gray, to mount a campaign that he could finance partly on his own, arguing he has at least an outside shot of taking out Paul.

Paul and his team are watching Gray closely, and his decision on whether to run -- expected by the state's filing deadline next week -- could impact the Kentucky Republican's calculation on how long to stay in the presidential race.

Gray and his team met this month with Sen. Jon Tester, head of the Democratic Senatorial Campaign Committee, and senior staff to discuss his interest in running. He has spoken with a number of Kentucky Democrats, including the lone Democrat in the congressional delegation, Rep. John Yarmuth. And the advice he's been getting seems to be universal: Run against Paul.

"I think at this point he's our strongest possible candidate, without question," Yarmuth said. "It sounded to me like it was something he really wanted to do."

Tester told CNN: "I think he can absolutely be competitive in Kentucky," adding that Gray is a "very, very appealing" candidate.

Gray and Paul both declined to be interviewed.

Even as he's been focused on his presidential run over the past year, Paul's team is preparing for the likelihood of a Senate reelection bid and leaving nothing to chance, according to several people involved in the discussions. They say that they are still holding out hope Paul pulls off a surprising comeback in the White House race. But they fully acknowledge that running a Senate race could cost well over $10 million, requiring an organization that takes time to build.

Paul's team is organizing campaign chairmen in all 120 counties, while reaching out to local GOP party leaders to ensure they'd be on board for the campaign. After holding 26 fundraisers in 2015, Paul's Senate campaign already has several on the books for 2016 as well, according to people involved in the effort. To reassure voters, Paul wrote two op-eds that ran in recent weeks in Louisville and Lexington newspapers, saying he is still "doing the job I was elected to do."

The campaign also is putting together a grassroots operation, including with students, similar to what powered him to an unlikely win in the 2010 Senate primary and later the general election. Two of his senior presidential campaign advisers, Chris LaCivita and Doug Stafford, will serve in similar capacities on the Senate run. The campaign headquarters will open soon in Louisville, with another office in Lexington.

In short, Paul's team is laying the groundwork for a run for his second term, even as the Kentucky Republican is publicly focusing on the upcoming Iowa and New Hampshire presidential primaries.

Paul spokeswoman Kelsey Cooper said that the senator has traveled "every corner of the Commonwealth" over the last year, while "listening to the concerns and issues of every Kentuckian." Despite his presidential campaign travel, Paul has done official and campaign events in Kentucky in every month except he did not hold fundraisers in November, aides said.

"Sen. Paul's No. 1 priority is doing the job he was elected to do," Cooper said. "And his nearly perfect attendance record in the Senate and strong presence in the Bluegrass is evidence of his unwavering commitment to all Kentuckians."

Democratic challenges

Paul has had the luxury of focusing on his presidential race since there's still no serious candidate against him for his Senate seat. Democrats had their eyes set on other potential Paul challengers, namely the former state auditor, Adam Edelen, but he decided against a run after losing his own reelection bid last year to remain as Kentucky's top financial watchdog.

Added to the hurdles for Democrats is the fact that Kentucky has become much redder in the Obama era, punctuated by Senate Majority Leader Mitch McConnell's trouncing of Democrat Alison Lundergan Grimes in his 2014 reelection bid and tea party favorite Matt Bevin's surprise win of the governor's office last year. Moreover, the marquee race in Kentucky this year will be whether Republicans can take power in the statehouse, a long-held goal of McConnell's, not necessarily the Senate race.

Nevertheless, a potential Gray candidacy is still attractive for Democrats. A wealthy executive of a construction company, Gray spent nearly $900,000 of his own money in his 2010 mayoral bid -- so he could likely give his Senate campaign a cash infusion, as well.

For that reason, Republicans are watching him closely.

"I think we'll win that seat," said Mississippi Sen. Roger Wicker, chairman of the National Republican Senatorial Committee.

Asked if a self-financing candidate is concerning, Wicker said: "It always is. It always is."

Moreover, Democrats said that Gray's progressive politics would energize the base. Gray, who is the state's first openly gay mayor, could bring out scores of more liberal voters in Louisville and Lexington, though he could have a harder time in more culturally conservative rural parts of Kentucky.

"He would energize the liberal base like probably no other candidate other than me," Yarmuth said. And Yarmuth, who said he has gotten assurances from Hillary Clinton's camp that she'd compete in Kentucky if she becomes the nominee, said that a big turnout in progressive parts of the state would help Gray.

While he wouldn't give Gray a "50 percent chance" of winning, he said, "it's not impossible."

Paul's calculation

Gray will have to make a decision by Tuesday, the state's filing deadline, something that could also influence Paul's timetable. Paul's advisers say his only focus right now is on the presidential race and nothing else. He will reassess depending on what happens in the early nominating contests.

Even if he loses handily in Iowa, Paul advisers expect him to continue onto New Hampshire, a state with a libertarian faction, which helped his father, Ron Paul, win nearly 11 percent of the vote in 2012.

Yet if Paul loses big in both states -- a distinct possibility if the current polls bear out -- and also has to worry about a Gray challenge back home, he'll have to make a decision on whether to proceed with a long-shot presidential bid ahead of Kentucky's own Republican caucuses on March 5.

The caucuses present their own dilemma. For years, Kentucky had held a primary to select a nominee. But Paul, seeking to circumvent a state ban on appearing on the ballot for two offices at the same time, forced through a change where Kentucky Republicans would hold caucuses instead to choose their nominee. He raised money for the effort and put his political capital on the line to convince skeptics, including McConnell, to back the plan to change the nominating contest to caucuses.

That presents Paul with an awkward dilemma: He could drop out before the caucuses that he was instrumental in creating; or he could compete in the caucuses and risk losing to one of the leading presidential candidates, just as he prepares to run for the Senate.

Paul's team says this is all speculation, since votes have yet to be cast in Iowa, which will kick off the nominating contest February 1.

Cooper, the Paul spokeswoman, said that the choice will be clear in a Paul Senate campaign, given the Kentuckian's work to battle the "liberal Obama agenda and war on coal."

"He looks forward to putting his record and his ideas up against anyone," Cooper said.

But Democrats are still holding out hope that they can broaden the map, forcing the GOP to play defense in a red state to help their campaign to regain control of the Senate.

"He absolutely can be competitive from my perspective," Tester said of Gray.