U.S. Sen. David Perdue's quest to overhaul the country's legal immigration system got a major boost this week after Senate Majority Leader Mitch McConnell indicated he was supportive of key aspects of the Georgia senator's plan.

The Kentucky Republican told Fox News' Laura Ingraham Wednesday that he agreed with the gambit Perdue and Republican colleague Tom Cotton have been pushing for months: that any congressional agreement giving legal status to so-called Dreamers who were brought to the U.S. illegally as children should be coupled with changes to the legal immigration system.

"I’m in favor of doing something on the DACA front," said McConnell, using the acronym for Deferred Action for Childhood Arrivals, the Obama-era program President Donald Trump said he'll end in March. "But I don’t think we ought to just do that."

McConnell said he would be interested in addressing two issues at the heart of Perdue and Cotton's bill: the federal diversity lottery and so-called chain migration.

"There are plenty of changes to the legal immigration system that should be added to any kind of DACA fix that we do," he said.

Perdue's bill, the Reforming American Immigration for Strong Economy Act, or RAISE Act, would eliminate the diversity lottery, a wildly popular visa program that has been in the spotlight due to its role in the recent New York City terror attack. The legislation also seeks to transition the legal immigration system to a merit-focused one, away from the current family-focused system that allows residents to bring members of their extended family to the U.S. Critics call that "chain migration."

McConnell is now the senior-most Republican on Capitol Hill to back key aspects of the legislation. That support could start paying dividends as soon as next week, when Congress must address government funding.

Democrats are insisting that a "clean" DACA fix -- without any other changes to immigration policy -- be included in a must-pass year-end spending deal. Republicans say changes to border security, immigration enforcement and potentially some components of the Perdue-Cotton bill should also be in the mix. But there is some division within the GOP about exactly how far it should go in overhauling the legal immigration system.

Bipartisan talks have so far stalled.

For his part, Perdue on Thursday called McConnell's backing a "milestone."

President Donald Trump endorsed the plan this summer, while Perdue's Georgia colleague Johnny Isakson has hedged.

Read more:  What's in Perdue's legal immigration overhaul?