McConnell's speech came a few hours ahead of a cloture vote on a motion to proceed on the Stop Student Loan Interest Rate Hike Act, S. 2343, which keeps the student loan interest rate on Stafford loans from doubling, from 3.4 percent to 6.8, for another year. Republicans are likely to employ a filibuster to block the bill moving forward.

Republicans, like Democrats, want to keep the rate at 3.4 percent, but they want to do it by taking money from a fund in the Obama administration's healthcare reform law to pay for the $6 billion extension. Democrats want to close tax loopholes on shareholders of S-corporations to pay for the extension. If Congress does not pass an extension by July 1, the rate is set to go up.

McConnell said the Democrats' legislation was really meant to win young voters before the Nov. 4 presidential election.

"What matters now for Democrats is that they find a way to drive a wedge between Republicans and a constituency that they’re looking to court ahead of the November elections. That’s what today’s vote is all about for them," McConnell continued.

The minority leader added that their legislation to extend the rate, which passed the House in April, was a "reasonable solution" for keeping the rate from doubling.

"If Democrats want to solve this problem, they should embrace it too — or at the very least offer a bipartisan solution of their own. The White House has done neither," McConnell said.