WASHINGTON — Top Senate Republicans pushed for an end to the stalemate over an emergency disaster relief package, with the leading Republican on the Senate Appropriations Committee warning the administration on Tuesday that failure to swiftly move forward was likely to derail major budget legislation needed in the coming months.

Senator Richard C. Shelby, Republican of Alabama and the chairman of the committee, pressed the need to resolve the issue on Tuesday with Vice President Mike Pence in a private lunch with Republican senators.

“My point was, if we can’t do this, what the heck can we do on something much bigger?” Mr. Shelby said, referring to looming deadlines to counter strict spending limits and avoid another government shutdown.

Negotiations over billions of dollars for recovery efforts have been gridlocked, primarily because lawmakers have struggled to reach a consensus over the amount of funding allocated for Puerto Rico. In their most recent offer, Senate Republicans had proposed an additional $300 million for the island, prompting top lawmakers to express optimism about a final deal.