Unite union chief Len McCluskey said Mr Corbyn and shadow chancellor John McDonnell were “not egomaniacs” and would not try to “cling on” at the head of the party.
Mr McCluskey said the leader and Mr McDonnell would “examine” the situation if Labour’s poll ratings remained “awful” a year before the scheduled election.
He told the Daily Mirror that Mr Corbyn should be given the time to prove himself as Labour leader.
But he added that the situation could change if Theresa May did not call an early election and Labour was still struggling in 2019.
“Let’s suppose we are not having a snap election. It buys into this question of what happens if we get to 2019 and opinion polls are still awful,” he said.
“The truth is everybody would examine that situation, including Jeremy Corbyn and John McDonnell.”
He added: “These two are not egomaniacs, they are not desperate to cling on to power for power’s sake.”
Mr McCluskey, who faces a challenge to his post as Unite general secretary from rival Gerard Coyne, warned that Labour needed to sort out its position on immigration and “get its narrative right on free movement” in order to avoid losing ground to Ukip.
He said: “We need to expose what Ukip stand for; not just anti-foreign but anti-workers’ rights, their hidden agenda on the NHS and welfare state.
“But it will only work if ordinary people believe Labour is listening to their concerns and has solutions.”
He said his position as one of Mr Corbyn’s strongest backers was “always based on critical support”.
“It’s my job to promote Unite’s policies, not Labour’s,” he said.
He claimed Mr Coyne was “being used by the right wing of Labour to try to have a proxy war about Corbyn”.
But Mr Coyne said: “I am astonished and deeply concerned that, at a time like this, Len McCluskey should deliver what amounts to a public ultimatum to the leader of the Labour Party.
“My criticism of his handling of the role of general secretary of Unite is not whether he has backed the right leader or the wrong leader of the Labour Party, but that he appears to think it is his job to be Labour’s puppet master.
“In 2015 and in 2016, he decided the Labour Party should be led by Jeremy Corbyn, and spent hundreds of thousands of pounds of Unite members’ money to make that happen.
“It is not in the interest of Unite’s members that the general secretary should spend so much of his time and their money playing politics.
“We are entering a momentous year. Article 50 will be triggered within less than three months, and the Government will enter into negotiations that will have an immense and lasting impact on the lives and working conditions of our members.
“Yet at this time of great insecurity for 1.4 million Unite members and their jobs, all Len McCluskey can talk about is the insecurity of Jeremy’s job.”