McCluskey, whose union is the largest in the UK and Ireland and a key backer of Labour, implied there was a risk it would no longer finance the party if it could not show that it is the "voice of ordinary working people".
Unite has given Labour £13.5 million since Ed Miliband became leader in 2010, according to the Daily Telegraph.
McCluskey said pressure was growing from Unite's members to "rethink" the relationship and that it was "essential that the correct leader emerges" in the Labour leadership contest.
“It is the challenge of the Labour party to demonstrate that they are the voice of ordinary working people, that they are the voice of organised labour," McCluskey told BBC Radio 5 live's Pienaar's Politics show.
"If they do that in a way that enthuses us, then I don’t believe that the mountain that’s ahead of us is un-climbable. But it’s up to them. If they don’t, if they kind of inject more disillusionment in the party then the pressure will grow from our members to rethink. It’s certainly already growing in Scotland.“
Asked about the Labour leadership contest, which has suffered two blows after Dan Jarvis said he wouldn't run and Chuka Umunna dropped out, McCluskey said: “It’s essential that the correct leader emerges, and that there’s a genuine debate about the direction we are going in.”
McCluskey said that some Unite members were calling for the union to be able to use funds for parties other than Labour - which would involve changing its rules.
“Very evidently Alex Salmond had occupied the left ground and the radical edge that Labour should have been occupying and ignored for years," he said. Nicola Sturgeon is an incredibly powerful politician and leader and is saying things which clearly struck a chord with ordinary workers in Scotland.”
Former Scottish Labour leader Jim Murphy
McCluskey's comments come after Labour's outgoing leader in Scotland, Jim Murphy, branded McCluskey the "kiss of death" as he announced his resignation yesterday.
Murphy urged UK Labour to distance itself from "the destructive behaviour" of McCluskey, saying: "The leader of the Scottish Labour Party doesn't serve at the grace of Len McCluskey, and the next leader of the UK Labour Party should not be picked by Len McCluskey," in a parting shot which captured high tensions with the union.
Labour leadership favourite Andy Burnham denied that he was the "union candidate" when he announced he would run to lead the party.
Mr McCluskey hit back at Murphy on 5live on Sunday, saying: “Obviously Jim is hurting and I can understand that, but he’s playing the same trick that right-wing media have played for a number of years, looking for a bogeyman as an excuse. I wasn’t the one that lost Scotland to the SNP...
“It wasn’t just Unite who called for him to step down... Yeah, he’s hurting at the moment, I understand that. I’ve never actually spoke to Jim, I’ve never had a conversation with Jim. I honestly haven’t.. and therefore it’s not personal. I said right at the beginning of the Scottish leadership election, I said it would be dangerous for Labour, in fact I think I said it would be a political death sentence, if Jim took it over.
"It wasn’t personal; it was because Jim has been at the epicentre of the ideology that has alienated Scottish working class for years and years and years.”
He added: “Scottish Labour has displayed an arrogance that, unfortunately, led us to where we were at the general election. So that’s why I thought Jim was the wrong person.”
It's not the first time McCluskey has threatened to sever ties with Labour. In April 2014 he told 5 live that the relationship could break down if Ed Miliband failed to embrace radical policies and lost the General Election.
The relationship between the party and the union has become increasingly rocky, and Murphy's comments made it clear that he saw McCluskey, personally, as the "problem" for Labour.
Mr Murphy said: "I know over the past few days I have been at the centre of a campaign by the London leadership of Unite the union, and they blame myself or the Scottish Labour Party for the defeat of the UK Labour Party in the general election.
"That is a grotesque insult to the Scottish Labour Party. It's a grotesque insult to our thousands of volunteers from someone who pays occasional fleeting visits to our great country.
"We have to draw the poison out of some of the personalities.
"Sometimes people see it as a badge of honour to have Mr McCluskey's support. I kind of see it as a kiss of death to be supported by that type of politics."
He added: "The Labour Party's problem is not the link with trade unions, or even the relationship with Unite members - far from it.
"It is the destructive behaviour of one high-profile trade unionist.
"One of the things about stepping down is that you can say things in public that so many people in the Labour Party only say in private.
"So whether it is in Scotland or in the contest to come in the UK, we cannot have our leaders selected or deselected by the grudges and grievances of one prominent man.
"The siren voice from behind a big desk in Unite's headquarters in London shouldn't be allowed to instruct what the Scottish Labour Party does.