A no confidence vote in the Lib Dem leader of Bristol City Council looks set to go ahead on 28 October.
The motion could leave the council without a leader
Labour still plans to table the vote over the appointment of councillor John Astley, who was later jailed.
They want Barbara Janke to apologise for selecting him and have accused her of "failing to respond positively to proposals to solve the impasse".
Ms Janke, who has condemned Astley's actions, told the BBC the Lib Dems never agreed to apologise.
Astley was convicted of election fraud and downloading child pornography in September.
Councillor Peter Hammond, leader of the Labour Group, said: "I provided her [Ms Janke] with three different possible statements which Labour and the Conservatives could have endorsed, bringing this sorry affair to a conclusion.
"I and Tory Leader Peter Abraham had a positive meeting with Barbara and I had hoped we could make progress on this.
"However the Lib Dem Chief Whip has informed me that their position has changed after an internal meeting and they are now unwilling to consider any apology whatsoever.
"It looks likely that Thursday's meeting to remove Barbara Janke as leader of the council will go ahead. "
Ms Janke said: "I invited Peter to this meeting and have been quite generous in moving this forward.
"Labour have been quite determined to say our selection procedure is flawed: it is not, and it is not so different from theirs.
"It is not a question of apologising, but of making sure this does not happen again and in moving forward."
She added that the Lib Dems have since "reviewed its
constituency party approval procedures; requested security measures for prevention of access to pornographic sites through council computers and requested enhanced CRB checks for all members".
Astley served as a Liberal Democrat councillor for the Lawrence Hill area of the city.
In September, he was sentenced to five months in prison after pleading guilty to 11 charges of electoral fraud.
He was already serving a 19-month jail sentence after pleading guilty to offences involving child pornography.
At the time, Ms Janke issued a statement saying: "We totally condemn his conduct."
Labour has 31 seats, the Liberal Democrats 28 and the Conservatives 11.