By Moira Warburton, Katharine Jackson and Gram Slattery

WASHINGTON (Reuters) - After tough negotiations to reach a tentative deal with the White House on the U.S. borrowing limit, the next challenge for House Speaker Kevin McCarthy is pushing it through the House, where it may be opposed by both hardline Republicans and progressive Democrats.

As Democratic and Republican negotiators iron out the final details of an agreement to suspend the federal government's $31.4 trillion debt ceiling in coming days, McCarthy may be forced to do some behind-the-scenes wrangling.

A failure by Congress to deal with its self-imposed debt ceiling before June 5 could trigger a default that would shake financial markets and send the United States into a deep recession.

Republicans control the House by 222-213, while Democrats control the Senate by 51-49. These margins mean that moderates from both sides will have to support the bill, as any compromise will almost definitely lose the support of the far left and far right wings of each party.

To win the speaker's gavel, McCarthy agreed to enable any single member to call for a vote to unseat him, which could lead to his ouster if he seeks to work with Democrats.

Hours before the deal was announced, some hardline Republicans balked at McCarthy cooperating with the White House.

"If Speaker's negotiators bring back in substance a clean debt limit increase ... one so large that it even protects Biden from the issue in the presidential ..., it's war," Representative Dan Bishop, a Freedom Caucus member, tweeted.

The deal does just that, sources briefed on it say: it suspends the debt ceiling until January 2025, after the November 2024 presidential election, in exchange for caps on spending and cuts in government programs.

Bishop and other hardline Republicans were sharply critical of early deal details that suggest Biden has pushed back successfully on several cost-cutting demands on Saturday, signaling McCarthy may have an issue getting votes.

"Utter capitulation in progress. By the side holding the cards," Bishop said.

Progressive Democrats in both chambers have said they would not support any deal that has additional work requirements. This deal does, sources say, adding work requirements to food aid for people aged 50 to 54.

The deal would boost spending on the military and veterans' care, and cap it for many discretionary domestic programs, according to sources familiar with the talks. But Republicans and Democrats will need to battle over which ones in the months to come, as the deal doesn't specify them.

Republicans have rejected Biden's proposed tax increases, and neither side has shown a willingness to take on the fast-growing health and retirement programs that will drive up debt sharply in the coming years.

Several credit-rating agencies have said they have put the United States on review for a possible downgrade, which would push up borrowing costs and undercut its standing as the backbone of the global financial system.

(Reporting by Moira Warburton in Washington; Editing by Heather Timmons and Kim Coghill)