Also on Friday, 60 members of the Democratic-controlled Senate signed on to a bill that would ensure troops were paid through a shutdown.
The move followed House Republicans' Thursday passage of a temporary resolution to fund the Pentagon and keep the government running for another week. That legislation, however, was dismissed by Democrats as political cover and ideological.
Reid and Boehner met with President Obama at the White House Thursday night for their fourth meeting this week.
Sources said the two sides agreed that their teams would find a number between $35 billion and $39 billion. The size of the cut would depend upon the composition of what was being cut. But Boehner would not commit to ending the demand that Planned Parenthood be defunded.
Boehner had Tea Party-aligned freshman who felt that if they couldn't meet the pledge they made of $100 billion in cuts to Obama's budget, they wanted to be able to present something to their constituents that they got in return -- ending government funding of abortion in Washington, D.C., for instance, or cutting off funding to Planned Parenthood.
"We've got to have something to go home with," Rep. Blake Farenthold, R-Texas, told ABC News. "If we don't get the full $100 billion, we need some of the policy riders. That way I can go back, look the folks at home in the eye and say 'Look, I couldn't get $100 billion but I got you X, Y, and Z."
Abortion took center stage in the fight over spending cuts. The abortion measure in the House Republicans' extension bill, and one they say they won't budge on, would reinstate a policy that prevented the District of Columbia from using locally generated taxes to provide financial help to poor women for abortions.
The House voted earlier this year to defund Planned Parenthood, but 41 Democrats in the Senate said they would not support that legislation. The White House said the president would not agree to any ban on funds to Planned Parenthood.
"We've come to realize that the moving target has now focused a bull's eye on women in America," Reid said. "We agreed on a number. But we are not -- we are not -- bending on women's health."
Planned Parenthood already is prohibited from using any federal funds for abortion-related services. Officials of the organization say more than 90 percent of what Planned Parenthood clinics do has nothing to do with abortion, but rather focuses on women's health services such as pap smears and mammograms.
Abortion opponents say federal funding for other services means money freed up for the purposes of conducting abortions, which they regard as ending human life.
The president cancelled his scheduled trip to Indiana today to remain in Washington, D.C., to take part in the negotiations.
It was unclear whether the Obamas would take their long-planned trip to colonial Williamsburg this weekend. All national parks would have closed in the event of a government shutdown.
The last time the government shutdown fully was in 1995, under President Clinton, for five days. A 21-day partial shutdown followed soon after.