GOP aides believe Democrats could be open to signing on so that both sides can punt the debate over how to deal with those across the board spending cuts into the late fall when broader budget negotiations will take place around a vote to raise the nation's debt limit.
Boehner's office is sending members home for the recess with talking points on the "strategy to end Obamacare" along with a PowerPoint presentation.
These describe the plan to hold more Obamacare votes designed to pick off Democratic support for the law, step up oversight about the administration's implementation of the law and ramp up communications efforts around the issue.
Mindful of the tense divisions and appetite from the Republican base for a high-profile fight about Obamacare, Boehner will continue to say publicly that all options are on the table.
After the meeting he repeated his public position to reporters, saying, "We've made no decisions about how we're going to proceed" on the funding bill.