"If it's about money, then we're there. We can do this deal," Matthews said. "We don't want this to go on to the next mayor."
Council is unlikely to hold the budget, at least not for long - if there's no budget by July 1, city employees, including D.C. 33 members, can't be paid.
"I don't necessarily know that the [D.C. 33] members would be too thrilled about that," Nutter said later in the day.
"We're always ready to sit at the negotiation table," the mayor added, but relations between the sides appear to remain strained.
A meeting scheduled for last Friday was canceled at the last minute.
On Thursday, Matthews and Nutter blamed each other for that.
Matthews said the sticking points continued to be furloughs, overtime, and pension reform, not salary.
"The city said, 'What's the point in meeting if we weren't going to move on those issues?' " Matthews said.
"We need proposals from District Council 33 that are responsive to offers we have put across the table," Nutter said. "Every other [municipal] union . . . has somehow figured out how to get to a negotiation table."
Matthews said the city has the money for a contract that would be in line with the deal given this year to the white-collar municipal workers who make up District Council 47.
The two unions historically have worked under similar contract terms.
Some Council members backed up Matthews' assertion.
"It is not a revenue issue," said Council President Darrell L. Clarke. "We have put in place the necessary revenues to give D.C. 47 and now 33 a contract."
And Councilman David Oh said money could have been set aside for the D.C. 33 contract.
"It's been there for a while and hasn't been used," Oh said. "If that's what it's for, I think it would be helpful if the mayor said that and did the contract."
Pressure also remains on Council to find money for the schools - a crowd of schoolchildren seated in the gallery before Thursday's meeting chanted, "Do your job, fund our schools."
Council did pass a bill Thursday that would raise $120 million for the School District by extending Philadelphia's extra 1 percent sales tax.
Five years ago, the extra tax was billed as a temporary measure to sustain the city through the recession. When the mayor signs the bill, as expected, the tax will become permanent.
Clarke is seeking the state's permission to split the proceeds of the sales tax - expected to be about $137 million next year - with the city's underfunded public employee pension system. But the bill Council passed Thursday guaranteed that the schools would get $120 million even if the state doesn't act on Clarke's request.
The School District is also counting on the $2-a-pack cigarette tax to help fill a $216 million budget deficit. That measure, however, would also require approval from Harrisburg.
Council also passed the city's capital-spending budget Thursday and could pass the operating budget at next week's meeting, now scheduled to be the last before the 12-week summer recess.
Council has few options left to get more money to the School District.
On Thursday, Clarke put aside a bill to establish the rate for the city's Use and Occupancy tax, which flows to the schools.
Increasing that tax, however, would require Council to add another meeting.
Asked Thursday why he held that bill, Clarke answered, "No particular reason."
Inquirer staff writer Jason Grant contributed to this article.