WASHINGTON — President Obama and House Republicans failed to reach agreement on a six-week extension of the nation’s borrowing authority during a meeting Thursday at the White House, but both sides agreed to keep talking and the Republican offer was seen as a first step toward ending the budget standoff. Twenty Republicans, led by Speaker John A. Boehner, went to the White House at Mr. Obama’s invitation after a day of fine-tuning their offer to increase the Treasury Department’s authority to borrow money to pay existing obligations through Nov. 22. In exchange, they sought the president’s commitment to negotiate a...