Council members in December narrowly approved the measure 5-4, saying it was the right thing to do.
Many citizens reacted angrily and the vote became one of the biggest issues in the recent campaign. Six of the seven council members elected April 1 pledged to eliminate the program, joining two holdovers who voted against it.
Five city workers and two hospital employees who signed up for the benefits will continue getting insurance coverage until Dec. 31 if the council ends the program.
"In all honesty I think we have more important things to get on the agenda, so let's get this over with and get onto other things," incoming Councilman Jerry Heimlicher said.
City human resources director Ann Crossey said partner benefits for the seven employees will cost taxpayers $6,700 this year.