Heated talks to decide who will control Worthing Borough Council are expected on Thursday.
Since the election, both Worthing's Liberal Democrats and Conservatives have 18 council seats.
Rules dictate that the town's mayor should have the deciding vote when there is a hung council.
As Worthing's mayor is Lib Dem Councillor James Doyle, the Lib Dems have proposed they should staff the executive and allow the Tories to chair six of the council's committees.
The move has enraged some Tory councillors who feel the power of the executive should be shared equally.
Re-elected Tory councillor George Stephens said: "The Lib Dems are not prepared to power-share.
"They are prepared only to throw us some little tidbits.
"The town has clearly specified, with 18 seats each, that it wishes us to share the responsibility for running the town."
Councillor Sheila Player, re-elected Lib Dem leader, said: "In the interests of sustaining and continuing strong political leadership and vision, we felt that the best split of the 12 posts would be for the executive to remain a united political group, while suggesting an equal number of posts and major responsibilities to the Conservative group.
"Although we could have retained total control by using the mayor's casting vote, we are happy to uphold the liberal principle of fair representation of votes.
"We hope the Conservatives will join us in a spirit of co-operation for the sake of Worthing and our people."
The Lib Dems' proposals will be discussed at the council's general purposes committee meeting on May 8 but final approval takes place on May 16.
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