Senate Majority Leader Mitch McConnell Addison (Mitch) Mitchell McConnellFEC flags McConnell campaign over suspected accounting errors Poll: 59 percent think president elected in November should name next Supreme Court justice Mark Kelly: Arizona Senate race winner should be sworn in 'promptly' MORE (R-Ky.) said a House-passed bill to provide a path to citizenship to nearly 2 million undocumented immigrants will "probably not" get a vote in the Senate.

McConnell indicated during an appearance on FOX News Radio on Wednesday that any legislation dealing with "Dreamers," immigrants who were brought to the country illegally as children, would have to be paired with broader changes to immigration.

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"I think the Dreamers have a sympathetic case," McConnell said. "There are circumstances under which I and others would be happy to support that. But we need to do more than that. You know, there's some genuine fixes on the legal immigration side and on the illegal immigration side that need to be addressed."

He added that he thought there "is a perfectly legitimate case for the Dreamers. ... But I think we need to do more than just that. And that's the context in which I would deal with that issue in the Senate."

McConnell's comments came after the House passed legislation on Wednesday to protect Dreamers and establish a path to citizenship for more than 2 million immigrants without legal status. Seven Republicans joined all Democrats in supporting the measure.

The White House Office of Management and Budget came out strongly against the bill, arguing that it "would incentivize and reward illegal immigration while ignoring and undermining key Administration immigration objectives and policy priorities, such as protecting our communities and defending our borders."

McConnell, asked about the prospects for a vote on the House bill, told Fox News that it would "probably not" come up in the GOP-controlled Senate. The immigration bill is one of several high-profile pieces of House legislation that have stalled in the Senate so far this year.

Attempts to pass a larger immigration bill would also likely face significant roadblocks on Capitol Hill, with the Senate rejecting four immigration bills last year.