Congress gave up Tuesday on a last-ditch bid to speed through emergency military aid to Ukraine before the end of the year, as negotiators failed to cement a deal that Republicans have demanded tying the money to a crackdown on migration across the U.S. border with Mexico.

“Our goal is as soon as we get back to get something done,” Senator Chuck Schumer, Democrat of New York and the majority leader, told reporters at the Capitol.

The delay punts the fate of Ukraine aid — and the complicated task of drafting new immigration laws — into early next year, when lawmakers will also face the daunting task of striking a broader spending agreement to avert a partial government shutdown by mid-January.

“We are not going to have a lot of time when we get back to get our security package and full-year funding bills across the finish line,” said Senator Patty Murray, Democrat of Washington and the chairwoman of the Appropriations Committee, admonishing Republicans to “get serious” about spending negotiations.

“January is not going to be an enjoyable month,” Senator Susan Collins of Maine, the top Republican on the committee, told reporters.

Senators have struggled for weeks to strike a bargain pairing about $50 billion in fresh security aid for Ukraine with border enforcement measures stringent enough to satisfy Republicans but not so severe as to alienate Democrats. They intensified their efforts late last week, after Mr. Schumer delayed the Senate’s holiday break to increase pressure for a deal before the new year.