The possibility that after weeks of stalled negotiations senators might be on the cusp of a breakthrough comes as President Obama and his top surrogates will begin on Monday their most aggressive push yet to rally Americans around his gun-control agenda.

Even though polls show that a universal background-check system is supported by nine in 10 Americans, the president has been unable to translate popular support for the measures into legislative momentum on Capitol Hill.

But in a move that could draw other Republicans as well as Democrats from conservative states who have not yet backed Obama’s agenda, Sen. Joe Manchin III (W.Va.), a key Democratic broker, has spent the past few days crafting the framework of a possible deal with Sen. Patrick J. Toomey (R-Pa.).