The offer comes on the 11th day of the government shutdown.
Below, more from the AP:
WASHINGTON — WASHINGTON (AP) — Officials say House Republicans are offering to pass legislation to avert a default and end the partial government shutdown as part of a package that includes cuts in benefit programs.
Senior aides to Speaker John Boehner and Majority Leader Eric Cantor outlined the proposal at a late-night White House meeting Thursday with senior administration officials.
In addition to ending the shutdown and increasing the debt limit, the proposal includes an easing of the across-the-board spending cuts that began taking effect a year ago, and replacing them with curbs in benefit programs that Obama himself has backed.
Among them is a plan to raise the cost of Medicare for better-off beneficiaries.
The officials spoke on condition of anonymity, saying they were not authorized to discuss private deliberations.
This story has been updated with more from the AP.