WASHINGTON—Senate negotiators moved closer to striking a broad deal on changes to U.S. border policy after the White House offered significant concessions to win Republicans’ support for separate legislation further funding Ukraine in its war against Russia. 

A deal would mark a breakthrough on an issue that has roiled Washington for decades, with the current talks driven by record numbers of migrants crossing the southern border and GOP lawmakers’ decision to tie new aid to Kyiv to changes in immigration policy. While Democrats have criticized that linkage, many have signaled in recent days they would be willing to make such a trade off.