WASHINGTON—Tightening the initial standard immigrants must meet when applying for asylum could form the basis of a bipartisan border agreement in the Senate, where a group of lawmakers is racing to strike a deal before the end of this year that could unlock billions of dollars in aid to Ukraine.

Senate Republicans have demanded a crackdown on asylum at the U.S.-Mexico border as a condition for backing President Biden’s request for emergency funding for Ukraine, part of a $106 billion proposal that also includes aid for Israel and Taiwan as well as money to bolster the immigration system. Senate Democrats have acknowledged that changes to border policy will likely be part of any deal.