WASHINGTON — With congressional approval of the latest emergency pandemic measure sealed on Thursday, the focus is quickly shifting to an escalating battle over whether Congress will provide hundreds of billions of dollars to states staggering under the costs of the coronavirus outbreak.

Anxious governors on the front lines of battling the pandemic have been clamoring for more federal help, saying their budgets are being stretched to the breaking point and their revenues are collapsing as they pour resources into health care while their economies are shut down. But the latest measure contained no new state aid, as Republicans resisted extending help even as state officials described their situations as increasingly dire.

“Many states are already reporting precipitous declines in revenues that fund state services in health care, education, public safety, transportation and other vital programs,” members of the National Governors Association wrote to congressional leaders this week. “States and local governments need robust support from the federal government as we navigate the response to this pandemic and to help foster the economic recovery that is ahead.”

Senator Mitch McConnell, Republican of Kentucky and the majority leader, alarmed and angered state officials on Wednesday when he said he wanted to approach the next round of pandemic legislation more deliberately. He said he was opposed to shipping money to state governments if they were going to apply it to fiscal problems unrelated to the pandemic, such as shoring up underfunded pension plans for public workers.