The vote ended a monthslong impasse over Obama's immigration policies.
Boehner told his conference the vote on funding DHS through September could be Tuesday.
McConnell worked with the minority leader to get the bill passed.
House Democrats said they expect the Speaker to bring a "clean" DHS bill to the floor.
Allegations date back to the 1990s.
Republicans in the upper chamber worry about a fight they consider a distraction.
The vote is a stunning defeat for Speaker John Boehner and his leadership team.
Foreign policy is quickly dominating the Conservative Political Action Conference.
Black lawmakers have entreated the House whip to attend next week's ceremonies.
Sen. Mark Kirk slams House for tying immigration language to funding bill.
The Speaker refused to say how the House would react to a clean funding bill.
Chatter of another coup attempt has grown louder.
The president said border agents will be "answerable to the head of Homeland Security."
The Senate vote on a clean bill funding DHS broke a weeks-long stalemate.
Sen. Harry Reid said Democrats will seek an "overwhelming vote" in favor.
“Until the Senate does something, we're in a wait-and-see mode,” Boehner said.
The Indiana governor might be running out of time to open the door on a presidential bid.
The Kentucky Republican is racing against the clock to fund DHS.