Senate Democrats had offered to consider GOP priorities that McConnell attached to the issue on Tuesday, in exchange for McConnell allowing a floor vote in the chamber for a standalone bill to raise the stimulus payments.
With McConnell refusing the proposal, which passed the House this week and is supported by President Donald Trump, it means the Senate can't vote on the prospect of giving struggling Americans more aid money.
A number of Republicans have also opposed the idea of increasing the amount, but McConnell is singlehandedly preventing the bill from even receiving an up-or-down vote.
McConnell first blocked the proposal on Tuesday, but said he would consider the raise as long as the Senate also considers two unrelated matters -- a provision to reduce legal protections for social media platforms and one creating an election integrity commission, both of which have been battles pushed by Trump in recent weeks.
Thursday, Schumer offered McConnell votes on both issues -- but the Republican leader still refused.
Schumer said allowing the Senate to vote on the standalone CASH Act -- rather than sandbagging it with the "poison pill" measures opposed by Democrats -- is the only way to get Americans "the $2,000 checks they need and deserve."
McConnell on Thursday dismissed the proposal to give Americans more money as "socialism for the rich."
Schumer answered, "Just give us a vote on the House-passed bill, and we can vote on whatever right-wing conspiracy theory you like."
Several other GOP senators have supported McConnell's position, but Republican South Carolina Sen. Lindsey Graham told Fox News Thursday that a standalone vote on the proposal "might pass" in the upper chamber. If it did, the raise would then only need Trump's signature to take effect.
Graham said he hopes the Senate will get another chance to consider the increase next week after the 117th Congress is seated on Sunday.
"I'd like a standalone vote in the new Congress on the $2,000 check," Graham said. "We have seven Republicans who've already said they would vote for it. We need five more. I think if we had the vote, we would get there."
With the ongoing debate in the Senate, Trump returned from his Florida resort to the White House on Thursday, ahead of schedule.