Sen. Jeff Flake of Arizona has considered running for his seat as an independent and given his blessing to an outside organization to conduct polling to explore the possibility, a source familiar with his thinking tells U.S. News.

Flake announced last Tuesday he would not seek re-election to a second Senate term next year in light of his dire chances of winning a Republican primary and his frustration with the tenure of President Donald Trump . Polling showed him trailing his insurgent challenger, Kelli Ward , by double digits.

But The Centrist Project, a group created to recruit independent candidates, approached Flake about running outside the dominant two-party system during a meeting on Friday. The senator signaled he was open to the possibility and interested in what polling would show, according to the source.

A spokesperson for Flake did not respond to an inquiry. After initial publication of this story, the senator tweeted that an independent run is "not gonna happen."

"I subscribe to the old saw Running as an Independent is the future ... and will always be the future," Flake said, continuing shortly afterward with another post : "I am a conservative and a Republican. This fever will someday break and we can rebuild our party once more."

The Centrist Project is planning to poll Flake's chances as an independent in a three-way race this week.

While Flake is skeptical about the feasibility of an independent run, the source says, he indicated he was open to the idea if he's provided a viable path to victory. Since Flake's announcement last week, a segment of GOP and independent-leaning donors also have been encouraging the option. The source attributed Flake's public disinterest in an independent run to public pressure from Republicans.

"Donors and like-minded Americans who care about a functioning democracy are realizing they're going to have to look to new options as party primaries continue to not only encourage but essentially guarantee the most extreme candidates," says Joel Searby, a strategist for The Centrist Project. "In the case of Jeff Flake, people from both the center-right and center-left realize that they can basically have whichever most extreme Republican emerges from the primary or, as an independent, a level-headed, principled Jeff Flake. When they start thinking in those terms, they begin to realize it's time for reconsidering the old ways."

If Flake should launch such a bid, it would further complicate what's already expected to be a highly competitive race. While Ward has been in the contest for a year, White House aides are leery about her candidacy and have been looking for alternatives.

GOP Rep. Martha McSally is seriously weighing entering the contest, indicating her interest to members of the Arizona congressional delegation last week, and Jay Heiler, a member of the Arizona Board of Regents, has formed an exploratory committee.

Democratic Rep. Kyrsten Sinema is already in the race and is not expected to face a difficult primary.