The contract was the subject of a Tuesday committee meeting.
Some council members suggested that more accountability and oversight are needed.
Councilman David James is asking the mayor to give council a seat at the table in an ongoing contract negotiation.
The 490 Projects says members of the public should be at the negotiating table.
The councilman believes the city can reopen them safely.
Council members are revisiting a "no confidence" resolution that asks the mayor for certain improvements.
Brian Allen Thurman, 49, was shot after officers pulled over a car that was reported as stolen around 10:30 p.m. Nov. 22 in the area of 21st and Gilligan streets, according to authorities.
The city says the impound lot has been at record capacity, and the Louisville Metro Police Department is working to make room for thousands of abandoned vehicles still on the road.
The stalled economy could also erase a potential surplus the city had projected.
The expected surplus could help the city dodge cuts in next year's budget.