Poirier stopped McGregor in the second round of Saturday's UFC 257 main event, avenging a loss to McGregor in 2014.
"He's already harassing everybody to get the rematch," Kavanagh told ESPN's Ariel Helwani. "We would love the rematch. I doubt that Dustin is against that.
"If we could get the rematch before the summer, that will be amazing. If it's not to be, well, then I guess I don't really know. Maybe he drifts off into boxing."
McGregor also has talked about boxing Manny Pacquiao this year.
McGregor was given a 180-day medical suspension by the ringside physician following the fight Saturday due to a potential injury to his right leg, according to MMA's official recordkeeper. However, McGregor can be cleared before that six-month window is up with a negative X-ray of his tibia and fibula, and Kavanagh indicated there didn't seem to be any structural damage.
There was one binding suspension given to McGregor, though: 45 days, the mandatory medical ban for fighters who are knocked out.
Khabib Nurmagomedov is currently listed as the UFC lightweight champ, but he has said he's retired and has given no indication that he'll change his mind. It's not certain how UFC president Dana White will handle the title picture, but it seems clear Poirier will be part of any scenario to crown a new champ. White indicated Poirier's next opponent could be Michael Chandler, who knocked out Dan Hooker in his UFC debut on Saturday's undercard.
"Dustin v. Conor 3 for the belt, in May or something like that; wouldn't that be amazing?" Kavanagh said. "We're 1-1. We gave him the rematch after the first win, and I hope he returns the sentiment."
Poirier sounded receptive Saturday night to a possible trilogy bout, but it might be a lot to ask for McGregor to fight for the title after a loss, even if it's for a vacant title.
Kavanagh also said McGregor is a couple of days away from "walking without any hindrance." Poirier landed several lower leg kicks, which had a huge impact.
"The leg is OK," Kavanagh said. "There's no serious damage. The peroneal nerve that runs down on that side of the leg, which was kicked a number of times, was completely dead, and then the leg started filling up with blood as well, so he couldn't move on it all.
"But as of today, he's in good form. He did a very tough workout this morning."