As the government nears a potentially devastating default, the White House and congressional leaders have begun bipartisan talks aimed at reaching a two-year budget deal, seizing on what could be their final chance at consensus before Speaker John A. Boehner’s exit ushers in what is expected to be more combative leadership in the House. Treasury Secretary Jacob J. Lew warned on Thursday that the United States would exhaust its ability to borrow on Nov. 5 if lawmakers refuse to increase the amount of money the government can legally borrow. That default date is weeks before lawmakers had expected, and it immediately increased the pressure to avoid a new fiscal crisis and break the gridlock that has become business as usual in Washington.