WASHINGTON — Congress returns to Washington this week under mounting pressure to strike an agreement with President Barack Obama on a bill to fund the government.

That’s proving difficult enough. Then comes the harder part for House Speaker John Boehner: convincing his many tea party-backed GOP freshmen the sort of split-the-differences measure Obama could sign isn’t a sellout.

Time is running short. Staff-level negotiations last week ran aground, and the principals are going to have to pick up the pace to have any chance of making an April 8 deadline to avoid a partial shutdown of the government. Right now it appears that the shutdown that both sides have sworn to avoid is possible — if not probable.

Republicans complain that the White House and Senate Democrats, after hinting of deep spending cuts, have been slow in officially offering them.

The frustration boiled over on Friday, with Republicans criticizing Democrats for not presenting significant cuts. An offer a few days earlier had ponied up just another $10 billion or so, GOP officials said, which prompted House Majority Leader Eric Cantor to accuse Democrats of “negotiating off of the status quo and refusing to offer any sort of serious plan for how to cut spending.”

The tough rhetoric was matched by volleys from Boehner, R-Ohio, and other House GOP leaders. That prompted Sen. Chuck Schumer, D-N.Y., to accuse Republicans of blowing up a near agreement on a “top line” of spending cuts that would have likely given Republicans more than half of their $60 billion-plus in reductions.

“After days of positive negotiations, with significant flexibility shown by the speaker, the House Republican leadership is back to agonizing over whether to give in to right-wing demands that they abandon any compromise on their extreme cuts,” Schumer said.

Democrats also say that House Republicans insist on using House-passed legislation slashing more than $60 billion from the current-year budget as the starting point for talks, pulling back from an agreement with Boehner’s office to work off a baseline essentially set at last year’s levels.

Boehner appears to be in a no-win situation. Any agreement with Obama is sure to incite a revolt among hard-line tea partiers who want the full roster of cuts and an end to funding for Obama’s signature health care law. And social conservatives are adamant that the measure cut off money for Planned Parenthood clinics that provide abortions in addition to the family planning services the government funds. Any attempts to outmuscle Obama with legislation that pleases tea partiers, however, would surely incite a shutdown.

“The speaker knows that when it comes to avoiding a shutdown, his problem is with the tea party, not Democrats,” Schumer said.