WASHINGTON—President Barack Obama and House Republicans began discussions Thursday on a GOP proposal to extend the nation's borrowing authority for six weeks, marking a new opening in the budget stalemate that risks a U.S. debt crisis.

At a White House meeting of the president and top House Republicans, Speaker John Boehner (R., Ohio) offered the proposal to extend U.S. borrowing authority in exchange for an agreement from Mr. Obama to negotiate on a broad range of budget issues.

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