WASHINGTON — After weeks of political haggling, the Senate agreed Tuesday to take up legislation that would give people running out of unemployment insurance benefits up to 20 more weeks of federal aid.

Senate Democrats, saying 7,000 people a day are exhausting their benefits, called on their colleagues to move quickly to a final vote. Republicans insisted they get a chance to offer amendments on the benefit bill and other issues.

Also in play was the possibility the bill would be used as a vehicle to extend another policy that has been central to the Obama administration's efforts to revive the economy: an $8,000 tax credit for first-time homebuyers.

The vote was 87-13 to bring the bill to the floor. Sixty votes were needed to pass that procedural hurdle.

The legislation would provide 14 weeks in extra financial aid for everyone exhausting their benefits by the end of the year, and another six weeks for those living in 27 states where the unemployment rate is at least 8.5 percent.