It comes after Mr Corbyn and deputy leader Tom Watson held separate talks with Mr McCluskey in an attempt to resolve the crisis at the top of the Labour Party.
They have sent a letter of support to the national Labour leader signed by Mayor Paul Dennett, and 28 of the 51 Labour councillors.
As the Conservative Party nears resolving the divisions caused by the European Union referendum, the impasse in the Labour party continues with embattled leader Jeremy Corbyn making it clear on Friday he will not resign.
"When we do things together, we're very strong".
"I'm convinced we shouldn't be unilateralist".
"I know 172 MPs sounds like a lot but most of those were from Blair's reign, who have been trying to get rid of Corbyn ever since he was elected".
"Angela Eagle - if you look at her constituency - her local constituency passed a motion in support of Jeremy Corbyn".
Eagle, who is tipped as a contender in a leadership contest, said: "The country doesn't have an effective opposition at the moment".
"That membership wants and expects all of us to work together in their interests".
"That would be a disaster for this country".
Labour MPs have lined up to condemn the threat to Kevin McKeever, a partner at Portland Communications and a former parliamentary candidate for the party, as "disgusting", "disgraceful" and "a betrayal for all we stand for".
"I'm not even convinced now he's a nice chap".
The controversial union baron was seen in Parliament between meetings and was said to be assessing the "lay of the land" in an attempt to broker a truce.
"That there's something rather more unpleasant lurking there".
Angela Eagle, the most senior member of the shadow cabinet to quit in the wave of resignations aimed at ousting the Labour leader, restated her view that Mr Corbyn "needs to go".
"Narrow majority pro-Corbyn but not as overwhelming as Momentum pretends - or would like".
Mr Corbyn was in support of Britain remaining in Europe, although he was criticised for his lukewarm support of the campaign.
Jeremy Corbyn seems to be going nowhere for now. "I'm carrying out that responsibility and I'm carrying on with that responsibility". "That is what I intend to do and I urge all my colleagues to do likewise".
Sources said union leaders had "reached out" to Mr Watson to see if they could find a negotiated settlement, with talks expected to take place on Tuesday morning.
Mr Corbyn, a socialist, says he retains the support of the party's grassroots members who will elect the leader in any contest, and that resigning would be a betrayal of the groundswell of support that saw him elected to the job last September.
But Mr Corbyn made clear he will not leave office, a source said.
Then it adds: "This ward calls on insert name of MP here to apologise for having voted for the war and calls on all other Labour MPs who voted in favour to do likewise".