In an effort to intensify pressure on Republicans — and address foreclosure and eviction moratoriums set to expire — House Democrats on Monday approved a stand-alone housing relief package. The provisions in the legislation were initially included in the $3 trillion stimulus bill that the House passed in May, but Senate Republicans have refused to take it up.

The legislation, championed by Representative Maxine Waters, Democrat of California and the House Financial Services Committee, would also allocate more funding to federal housing programs, provide $11.5 billion for homeless assistance grants and provide billions of dollars to help cover rent, mortgage payments and utility bills.

But in the Senate, Republicans blocked an effort to approve another package of state and local funding, as Democrats renewed calls to begin negotiating another pandemic relief package amid a continuing surge in cases. A number of existing relief measures, including a $600 expanded unemployment benefit, will expire soon without congressional action.

Democrats tried to use a procedural maneuver that would allow the legislation to pass the Senate without the full chamber present, but Senator Rick Scott, Republican of Florida, objected, citing concerns that the national debt and deficit would reach “unsustainable levels.”

The dispute came as Speaker Nancy Pelosi of California and Senator Chuck Schumer of New York, the minority leader, slammed Senator Mitch McConnell of Kentucky, the majority leader, for his unwillingness to begin another round of talks. They wrote in a letter to Mr. McConnell that “now is the time for action, not continued delays and political posturing.”

Though there is widespread acknowledgment on Capitol Hill that another relief package is needed, Mr. McConnell and top Senate Republicans have pushed to delay any negotiations or legislation until after the chamber returns from a two-week July 4 recess.

Republicans, however, have repeatedly stressed that Congress should wait before doling out another round of taxpayer aid so they could see the impact of the $2.2 trillion stimulus package, which became law in March.

Several economists have pleaded with Congress to continue spending and maintain the unemployment extension, but Republicans remain divided over how to balance the need for economic relief with calls to rein in spending.