Even as Congress moved toward extending some or all of the tax cuts, the commission was facing a potential break between its elected officials and its members who do not have to face the voters, people familiar with the panel’s deliberations said.

The panel needs the agreement of 14 of its 18 members for its recommendations to go to Congress for a vote, a prospect that has been a long shot since it convened earlier this year.

But a failure to win support from at least some lawmakers from both parties would undercut the panel’s ability even to set the terms of a debate between the administration and Congress over how to proceed in addressing chronic annual budget deficits and a national debt that it mounting toward unsustainable levels.