The critical deadline for the Senate-based negotiations is Sunday afternoon, according to Reid and McConnell. If they can come up with something by then, the Senate could approve it by late Sunday or late Monday morning, giving the House the rest of New Year's Eve to consider it.
Even under this optimistic timeline, Senate and House leaders would need cooperation from the hard-liners in their respective caucuses, who could turn to procedure to block a compromise.
Under normal Senate rules, for instance, a new piece of legislation could take a full week to begin consideration, have full debate and then final passage.
To avoid that, the Senate would instead likely take up a bill that has already passed the House — and which keeps all taxes at their current rates — and gut it and replace it with new language. Even so, McConnell would need to ensure that the most conservative Republicans don't filibuster the measure or then force a lengthy 30-hour debate, which is allowed under Senate rules.
If all of that can be worked out, the Senate would ship a bipartisan deal across the Capitol Rotunda by Sunday night or Monday morning.
The first thing that would have to happen in the House is an agreement to waive a rule instituted under Speaker John Boehner, R-Ohio, that requires a full day between the time a bill is published and when it is voted on.
Boehner has also discussed amending a Senate-passed bill with a more conservative tilt on taxes — a possibility he again raised during Friday's White House meeting.
"Let us know what you come up with, and we'll consider it — accept or amend it," Boehner told the other leaders, according to the notes of a Boehner aide.
Amending the bill would send it back across the Senate for last-minute consideration, with many of the same procedural hurdles applying again and with the clock ticking steadily toward midnight.