Burnham made the comment when asked whether it was time for a woman to lead Labour at BBC Radio 5 Live’s leadership hustings.
He quickly clarified his remarks, saying: “When the right candidate comes forward. It could be now, it could be in the future. Who knows?”
However, Diana Johnson, the MP for Kingston upon Hull North, who is supporting Cooper, questioned whether he was suggesting the party was not ready for a female leader.
“What on earth does Andy mean by saying he’d support a woman leader when the time is right? Is he suggesting that even now – in 2015 – the Labour party is not ready for a woman leader? After 100 years of campaigning for women’s equality are we really saying we don’t think a woman can do the top job? I think the big question to Andy is:If not now, then when?”
One of Cooper’s campaign aides also tweeted: “When is ‘the right time’ for a woman leader? 5yrs, 10yrs? Terrible message to send out to young women and girls.”
Another Cooper supporter, Helen Goodman, the MP for Bishop Auckland, told the London Evening Standard: “Andy must be a) out of touch, b) very tired or else c) very sexist. To be fair, I think on balance it is either a or b.”
A spokesman for Burnham said: “There’s only one thing that matters and that is which candidate has the right policies to beat the Tories.”
His camp has previously been accused of sexism by Cooper supporters after a source close to his camp said her tactics were “straight out of the Ed Balls playbook” – in a suggestion that she was being influenced by her husband, the former shadow chancellor.
At the time, Cooper said this seemed to be “going back almost to the days of the suffragettes … [when it was said] they couldn’t do things because of things their husbands have done”.
Cooper’s camp previously claimed Burnham’s campaign was engaging in “old-style bullying from the boys”, after the suggestion that Cooper and Liz Kendall should take a step back and let him become the main rival to the frontrunner, Jeremy Corbyn.
Responding to that, Lucy Powell, a senior Labour MP backing Burnham and former campaign chief for Ed Miliband, said: “This contest is way beyond gender politics. We are in a fight for the future of our party and country. Only Andy can beat Jeremy and to suggest otherwise is frankly misleading.”
guardian.co.uk © Guardian News and Media Limited 2010