WASHINGTON—Top Senate leaders said they were within striking distance of an agreement Monday to reopen the federal government and defuse a looming debt crisis just days before the U.S. could run out of money to pay its bills.

Senate Majority Leader Harry Reid (D., Nev.) said on the Senate floor that the leaders had made "tremendous progress" toward a deal and that he was hopeful Tuesday would be a "bright day." The Senate's Republican leader, Mitch McConnell of Kentucky, seconded Mr. Reid's optimism. "We've...