WASHINGTON — The one sliver of real hope came a week ago, in the darkened Capitol on the Sunday night after Veterans Day.

Called away from dinner tables, the Jets-Patriots game on television and, in one case, a soccer team party, several Democratic members of the special Congressional committee on deficit reduction raced to the office of Senator Patty Murray for a hasty 8:30 meeting to discuss the outlines of a potential agreement. Crucially, it appeared to have the backing of at least one Republican on the 12-member panel even though it included a tax increase.

As the members spoke, they began to see the outlines of a deal, tentatively agreeing on tax rates, revenues, spending cuts and changes to Social Security and Medicare, according to interviews with members of the committee and their aides.

Whether the committee could reach the finish line was very much in question, but at least it was in sight, for the first time in the nine weeks the group had been meeting.