Polar Plungers take an icy dip in Lake Michigan in 2013’s event. | Sun-Times photo

Jimmy Fallon says Chicago Mayor Rahm Emanuel has an open invitation to be a guest on “The Tonight Show.”

“I don’t mess with that guy,” Fallon said in a Monday Sun-Times story, where he played up Rahm’s reputation as a guy who gets what he wants — or else. “All he has to do is make a phone call. He can host if he wants to. I’m scared of that guy.”

On Wednesday, Emanuel took to Twitter to extend an invite of his own to the new host of NBC’s late-night institution:

It’s on.

Emanuel’s press secretary Tarrah Cooper confirmed Wednesday that the mayor promises to be on Fallon’s show if Fallon joins him and some other brave souls at North Avenue Beach the morning of Sunday, March 2, for a refreshing dip in icy Lake Michigan.

Emanuel agreed to do the Polar Plunge, a fundraiser for Special Olympics Chicago, because local kids met a goal last summer of reading 2.1 million books as part of the citywide Chicago Summer of Learning program.

Fallon responded to Rahm’s throwdown on Thursday:

Rahm’s camp is amping up the pressure:

Who knows? This might just happen. After all, Fallon let Harrison Ford pierce his ear on “Late Night.” The Polar Plunge might be less painful.