(CNN) As Tim Pawlenty weighs whether to run for Senate in Minnesota, Republicans have been attempting to persuade the former governor to jump into the race -- with outreach by a roster party leaders, donors and Republican activists, according to multiple sources familiar with the discussions.

The Senate seat has emerged as an unexpected pickup opportunity for Republicans in a potentially punishing midterm election year after Sen. Al Franken's abrupt resignation following sexual misconduct allegations.

Franken's replacement, Democrat Tina Smith, previously served as lieutenant governor but does not have an outsized profile in the state.

Pawlenty, meanwhile, was elected governor in 2006 and a candidate in the 2012 Republican presidential primary and in contention to be Mitt Romney's running mate

One vocal advocate for Pawlenty's candidacy has been former Sen. Norm Coleman, who sat down with Pawlenty multiple times before Christmas to pitch him on the race, both in Minnesota and Washington.

"He is certainly, from my perspective, the ideal candidate," Coleman told CNN. "After the Franken embarrassment, he's someone people know and trust."

Pawlenty, through an intermediary, declined to comment.

Recruitment efforts could be complicated by stiff political headwinds facing Republicans in the upcoming election. A CNN poll last month found Democrats with a staggering 18-point advantage over Republicans nationwide in a generic ballot.

"This is a challenging environment, but Tim has never been afraid to face challenges," said Coleman. "Fear is not the deciding factor for him. He's got enough going that he could easily overcome the electoral dynamics, the national trends."

Indeed, Pawlenty managed to narrowly win reelection to the governor's mansion in 2006, an election cycle that pummeled Republicans across the country.

That record, plus his strong name I.D. in Minnesota and deep connections among party donors, have caused Washington Republicans to zero in on Pawlenty as a prize recruit. Some sitting senators, including members of the Republican leadership, have reached out to the former governor in recent weeks to urge him to run, multiple sources said, as have prominent party donors.

The White House also believes Pawlenty is the potential candidate who is best positioned to pick up the seat for Republicans, said one party strategist who works with the administration. Pawlenty remains close with his former presidential campaign manager, Nick Ayers, who is now chief of staff to Vice President Mike Pence and an influential voice in the administration on political decisions.

Pawlenty said in a recent interview on CNN's "New Day" that he remains "politically retired" for now. But he had already been weighing a return to politics before Franken stepped down, as a potential candidate in the race for governor. His confidantes acknowledge that Pawlenty has more experience serving in executive roles, including as governor and more recently as CEO of Financial Services Roundtable.

Pawlenty was close to running for Senate once before, in 2002 midterm election cycle -- but then-Vice President Dick Cheney convinced him to exit the race just as Pawlenty was prepared to announce his bid. The last-minute intervention cleared the way for Coleman in the Senate primary, and Pawlenty decided to run for governor instead.

"We were all so much younger then," Coleman said, adding that a Senate seat might not hold the same appeal for Pawlenty today. "There's a call to service, and I believe Tim's open to service. But then the question is, what does he want to do?"

Former Rep. Vin Weber had "a long talk" with Pawlenty prior to Franken's resignation, urging him to run for governor, "and I think he was very interested in that," Weber told CNN. Although Weber would now support Pawlenty for governor or Senate, "they're totally different races" -- with a Senate bid likely presenting greater political challenges for a Republican running in 2018.

"I have no doubt that (Pawlenty) would be more excited about being in public life than about being in private life," Weber said, "but that's not all there is to the decision."

Former Minnesota Republican Gov. Tim Pawlenty maintains he is "politically retired" when asked about a bid for Sen. Al Franken's seat https://t.co/4sToWDeRU2 — CNN Politics (@CNNPolitics) December 12, 2017

Who else is running?

Other potential candidates have been expressing interest in the Senate contest.

Karin Housley, a Republican state senator, has already announced her intention to jump in the race.

Former Rep. Michele Bachmann, in a recent interview, said she has been urged to run for Senate, but has not reached a decision.

Bachmann, a one-time Tea Party star, might be one of only a few Minnesota Republicans who could hope to match Pawlenty's name recognition in the state. But Republicans believe the former governor could uniquely marshal the resources and broad support needed for a competitive campaign.

Those close to Pawlenty believe he will make a decision soon, in order to begin putting the pieces in place for a campaign.

"The clock is ticking to run a statewide campaign," Coleman said, noting fundraising pressures in particular.

"I've laid out positives of why (Pawlenty) would be the right candidate at the right time," Coleman added. "He'll have to make some choices."