The plan is seen as an attempt to undermine Mr Watson, the Labour MP for West Bromwich East, who has been caught up in Labour’s internal battles since Jeremy Corbyn became Labour leader.
Mr McCluskey told BBC Radio 4: “I think there’s a debate at the moment that says it would be good to have two deputies. One male, one female.”
Asked if he personally backed such a change, the Unite General Secretary said: “That sounds like a good idea to me.”
Mr McCluskey’s position has head of the UK’s largest trade union appears secure after he saw off a leadership bid from Gerard Coyne, the union’s former West Midlands Regional Organiser.
The battle for control of Unite grew increasingly bitter, with Mr McCluskey complaining that Mr Watson and Sion Simon, a West Midlands Labour MEP and former mayoral candidate, were supporting his rival.
Mr McCluskey also said he was prepared to break the law in order to launch strike action against the public sector pay cap, saying he was ready to defy a legal requirement that strike action needs to be approved by a ballot with a turnout of more than 50%.
He told the BBC: “We will always stand shoulder to shoulder with our members.
“If the Government have pushed us outside the law then they will have to stand the consequences.”
Asked if he was prepared to back a strike called on a turnout of less than 50%, Mr McCluskey said: “If they haven’t managed to hit an artificial threshold that this Government have foolishly put onto the statute books, then I will stand by our members and we will all live, including the Government, we will all live with the consequences of that.”
Labour’s shadow justice secretary, and shadow Lord Chancellor, Richard Burgon repeatedly refused to say whether he would back a strike called without a 50% ballot turnout.
Asked if he would support an illegal strike, Mr Burgon told the BBC the question was “hypothetical”.
He said: “What a Labour government would do would be to repeal the Trade Union Act which is seeking to stop trade unions from taking action to stop ordinary people... from suffering a 14% pay cut in real terms.”
The current Labour leadership has a position of supporting strikes, in contrast to recent previous leaders who usually attempted to avoid being seen to support industrial action.
Labour Shadow Chancellor John McDonnell backed the strike by Birmingham refuse workers, when he spoke at a rally in Brighton where trade union body the TUC is holding its annual congress.
He also referred to claims that senior council managers had blocked a plan drawn up by council leader John Clancy which might have ended the bin strike - and urged councillors to make their own decisions.
Mr McDonnell said: “I do not see your strike as strike against Labour. I see your strike as a strike against austerity. Simple as that.
“Our advice to Labour councillors wherever, when they go into dispute, is to get round the table.
“And insist on the fact that it isn’t chief executives or officers of the council that determine the future of a council’s policies, but the councillors themselves.
“So we want negotiations to resolve this dispute immediately.