And it had worked before, when Boehner and Obama agreed to a deal for fiscal 2011 spending. Taking their cue from Obama, Senate Democrats held their nose and approved it.
By late Friday, it appeared that Obama was not able to offer enough in the way of spending cuts, so Boehner dropped those talks and turned back to the Senate to find a way forward. This implies a less grand agreement, possibly one that fixes the debt ceiling problem for a shorter period of time.
And that's if a deal can be done at all. Senate Democrats continue to oppose significant spending cuts and prefer tax increases.
All of this is subject to change, especially after a planned Saturday morning White House meeting involving all participants.
If that doesn't move the issue along, one voice that might quickly help Boehner choose a dance partner is Wall Street. The financial markets so far have been patient, but could become agitated this week if it looks like a default or triage-style federal spending is possible.
At the same time, market disruptions could end up being the push both sides need to find an excuse for backing away from their dug in positions. A nervous Wall Street might allow Republicans and Democrats to point to the greater need of ensuring financial stability and allow them to reach a deal that cuts spending more than Democrats prefer and raises new taxes more than Republicans want.
On the floor next week, the House has a full schedule. It plans to take up legislation to extend the term of FBI Director Robert Mueller, consider the Interior, Environment and Related Agencies Appropriations Act.
The House may also take up a balanced-budget amendment.
Below is a more detailed look at the week ahead:
The House meets at noon for legislative work, and is expected to consider three suspension bills. These are S. 1103, to extend FBI Director Mueller's term, H.R. 440, to establish a religious freedom envoy in Asia, and H.R. 1383, the Restoring the GI Bill Fairness Act, as amended by the Senate.
The House will also start work on H.R. 2584, the 2012 Interior, Environment and Related Agencies Appropriations Act.
The Senate meets at 2 p.m. and is in session for the rest of the week, but does not have a set schedule. The Senate adjourned Friday without taking up an FAA extension, so work on that issue is possible.
The House meets at noon for legislative work, and is expected to continue work on the Interior Department spending bill.
At some point, the House will take up eight suspension bills to name post offices. These bills are H.R. 2548, 2244, 2213, 789, 1975, 1843, 2062, and 2149.
The official picture of the 112th Congress will be taken on Tuesday during the first series of votes.
The House may take up a balanced budget amendment on these days, or other bills related to the debt crisis.
Work is also possible on H.R. 2587, which would prohibit the National Labor Relations Board from dictating employment decisions to companies.
The House meets at 9 a.m. and will likely work to complete the bills listed above.
Legislative work on bills related to the debt ceiling is possible over the weekend.