The Democratic push for more federal aid to states and cities contending with a budget crunch because of the pandemic was a major sticking point in the latest round of legislation to respond to the outbreak, slated to clear Congress on Thursday, with Senate Republicans refusing to budge on the issue in negotiations.

Mr. McConnell had already made it clear on Tuesday that he would approach the next round of emergency funding much more cautiously. He went further in his comments on Wednesday, saying that he did not want Washington rescuing the pension plans of states that were struggling to keep up with their commitments to an array of union workers.

“We’ll certainly insist that anything we’d borrow to send down to the states is not spent on solving problems that they created for themselves over the years with their pension programs,” he said.

Mr. McConnell is likely to run into considerable resistance to his position from the White House, congressional Democrats, governors of both parties and even some lawmakers in his own party.

After the Senate vote Tuesday, top Democrats said that President Trump was open to the idea of helping states with their pension issues and that Mr. McConnell had been the chief obstacle to getting money for local governments this time around.

In a pair of Twitter posts on Tuesday, Mr. Trump said that after he signed the latest round of emergency pandemic legislation, he was ready to open discussions on help for “State/Local Governments for lost revenues from COVID 19, much needed Infrastructure Investments for Bridges, Tunnels, Broadband, Tax Incentives for Restaurants, Entertainment, Sports, and Payroll Tax Cuts to increase Economic Growth.”

Democrats were also confident that Congress would move forward with more relief for states, with Democratic leaders of the House and Senate promising a robust program. Several Republican lawmakers have introduced or signed on to proposals to provide huge sums to struggling states.