The deal under consideration would increase funding for discretionary spending on military and veterans while essentially holding non-defense discretionary spending at current year levels, said the official, who requested anonymity because they are not authorized to speak about internal discussions.
The defense and veteran affairs funding matches Biden's budget released earlier this year, a second U.S. official said.
Fast-growing health and retirement programs would not be affected, even though they are projected to push U.S. debt levels higher in coming years.
Republicans have rejected Biden's proposed tax increases on corporations and wealthy people, while Biden has resisted Republican proposals to stiffen work requirements in some antipoverty programs and loosen oil and gas drilling rules.
A two year extension would mean Congress would not need to address the limit again until after the 2024 presidential election.
However, items including work requirements for recipients of federal aid were still holding up the deal, an official said.
Republican negotiator Representative Garret Graves said late Thursday that the White House is "refusing to negotiate on work requirements," which he called "crazy." He said disagreements over funding social security and Medicare versus work requirements continues to be an issue between the two sides.
Also, the agreement presently leaves many details to be sorted out in the weeks and months ahead.
Each side in the negotiations will have to persuade enough members of their party in the narrowly divided Congress to vote for any eventual deal, no small feat with far-right Republicans saying they wouldn't back any deal without sweeping spending cuts - and progressive Democrats resisting new work requirements on anti-poverty programs.
"The only way to move forward is with a bipartisan agreement. And I believe we will come to an agreement that allows us to move forward and that protects the hardworking Americans of this country," Biden said on Thursday.
A failure by Congress to raise its self-imposed debt ceiling in the coming week could trigger a default that would shake financial markets and send the United States into a deep recession.
The standoff has unnerved investors, pushing the government's borrowing costs up by $80 million so far, according to Deputy Treasury Secretary Wally Adeyemo.
Several credit-rating agencies have said they have put the United States on review for a possible downgrade, which would push up borrowing costs even further.