Senate Majority Leader Mitch McConnell, R-Ky., said he expects Republicans and Democrats to strike a compromise deal to allow recipients of the Deferred Action for Childhood Arrivals program to remain in the U.S. in exchange for measures on border security or to end "chain migration," which makes it easier for family members of current U.S. residents to enter the country.

McConnell said in an interview that aired Saturday that he expects a deal to be struck that improves the legal immigration system and laid out multiple possibilities.

"Yeah I think so," McConnell told MSNBC's Hugh Hewitt, when asked if there's room for a compromise on the issue. "The Democrats desperately want DACA. Republicans are not necessarily opposed to that. The president has set it up in such a way that we have an incentive to act here because he's given us six months to come up with a proposal."

"I think there ought to be something relate to making the American legal immigration system better, achieved along with DACA," McConnell said. "It could be border security, it could be chain migration, it could be the diversity quotas, but something that tangibly improves the legal immigration system in this country ought to be attached to DACA."

On Thursday, President Trump stressed to a group of Republican senators that he does not want a DACA fix to be part of a year-end spending deal as the current one expires on Dec. 8. Sen. John Cornyn, R-Texas, the No. 2-ranking Senate Republican, said he doesn't expect any fix to take place until February.

McConnell did not put a timeline on any deal.

One impetus behind possible immigration discussions — outside of the March deadline for DACA — is the terror attack in New York on Tuesday that killed eight individuals. President Trump has been outspoken since the attack, saying he wants to "step up" the administration's "extreme vetting" and called for a merit-based legal immigration system. McConnell agreed that a discussion is needed.

"The president's been talking about improving our vetting for some time. You've got to look at a guy like this and say, 'How'd he get here in the first place?'" McConnell said. "And so I do think it's worth taking a look at."