Boston City Council president Michael Ross says he’ll reject the firefighters’ new contract unless the union gives the city a “significant concession” - an 11th hour request the union brushed aside.

“If we lost the arbitration, would the mayor throw out the arbitrator’s award and go back to bargaining with us? I think not,” Boston Firefighters Local 718 president Ed Kelly told the Herald. “We played by the rules.”

But Ross, drawing from recommendations to the council from an MIT labor relations expert, called on Local 718 to voluntarily head back to the negotiating table.

“In the end, I believe (the contract) is too much, given our challenges,” Ross said. “I cannot vote for it unless the union does what unions are doing around the country, and that’s recognize the fiscal challenge we are in and give a significant concession.”

Mayor Thomas M. Menino yesterday sent the council a request to appropriate $18 million to fund 2010 raises in the four-year contract. The mayor says the deal will cost the city $74 million.

Asked if the mayor would be open to going back to the table with the union, Menino spokeswoman Dot Joyce said: “This has gone on a long time, but he has to support this by law.”

The state arbitrator’s award gives raises similar to those given to other unions, except for an extra 2.5 percent hike this year that was included as a concession for random drug and alcohol testing. Other unions have also been rewarded for drug testing.

While the arbitration is binding, Ross urged the union to consider bending to help ease the city’s financial woes.

‘It’s not the (Menino) administration asking (the union) to do this. It’s for the citizens of this city who are beside themselves on the size of this contract and the notion that they have to pay for a firefighter to show up sober for work,” Ross said.

Kelly denied the union has the authority to tinker with the contract, saying: “We are bound by law to accept this award.”

He noted that he would welcome the council into future negotiations.

“Maybe going forward the city council needs to be involved in the collective bargaining process. Maybe that’s a viable avenue that could mend these dysfunctional relationships,” Kelly said.

Ross joins Councilor Chuck Turner in pledging to vote the deal down. Also yesterday, Turner called for an audit of the city’s multiple bank accounts in response to Local 718 claims that the city has understated its available cash.