WASHINGTON—Senate leaders attempting to avoid a U.S. debt default remained at loggerheads Sunday and escalated the standoff by reopening the contentious issue of automatic spending cuts, damping hopes that some of Congress's most canny negotiators would break the impasse.

As the search for a way to end the partial federal shutdown and avoid a debt crisis shifted to the Senate, Democrats made plain that one of their top priorities was to diminish the next round of across-the-board spending cuts, known as the sequester, due...