Charles Pugh said the concessions will be big enough to avoid a state-appointed emergency manager, The Detroit Free Press reported.
"They said they feel it's going well and they expect it to wrap up in the next two days," Pugh said. "I'll call them this evening to see what movement they have made. I think they have made movement in healthcare, which is a big chunk."
Michigan Gov. Rick Snyder has said the city must wrest $105 million in concessions from its unions, the Detroit Free Press reported. If not, an emergency manager would take over city operations.
In addition, a recent report said the city has a cash-flow problem that could leave it without money for operating expenses by April 1, the News said.
This week, the council discussed possible cuts, including raising bus fares from $1.50 to $2, closing recreational centers and ending subsidies to cultural institutions like the famed Detroit Institute of Arts.
Mayor David Bing has proposed eliminating 1,000 city jobs and said another 1,300 employees could be laid off unless unions agreed to concessions.
Chipotle plans first price increase in 3 years
Putin thinks Obama would save him if he were drowning