Whether it is a serious reconsideration, or just chatter with colleagues is the subject of some disagreement. There are also conflicting accounts of whether Corker has initiated the conversations, or whether he has had them with colleagues who are pushing him to think again.

But several sources say the issue has come up in recent conversations Corker has had with fellow Tennessean Sen. Lamar Alexander and with South Carolina Sen. Lindsey Graham, two of Corker’s friends.

It also came up at least once in a conversation with Majority Leader Mitch McConnell, CNN is told, posing a bit of a dilemma.

McConnell likes Corker, and would have preferred that he sought re-election. But once Corker announced he was not running, the GOP establishment quickly rallied behind Rep. Marsha Blackburn, who is already in the Senate race and has support across the party’s often fractious spectrum.