In an overt dig at her only male rival, Ms Phillips, a prominent backbencher, said men should sometimes “pass the mic” if they truly believe in gender equality.
The Birmingham Yardley MP said Labour had a “blind spot” when it comes to electing women leaders – an omission that proved particularly embarrassing for the party when the Conservatives elected Theresa May as prime minister.
Her comments came after Sir Keir cemented an early lead in the five-way contest, with a poll of Labour members putting him well ahead of his nearest rival, the left-wing candidate Rebecca Long-Bailey.
Ms Phillips, who is polling behind both candidates, used the first leadership hustings on Saturday to accuse candidates from the shadow cabinet – including Ms Long-Bailey, Sir Keir and Emily Thornberry – of keeping quiet over antisemitism during Mr Corbyn’s leadership.
Asked if it would be embarrassing to have another male leader, Ms Phillips told Sky News’ Ridge on Sunday: “I think it will be embarrassing and what’s more, it gives absolute grist to the mill and ammunition to our other side.
“I have had Tory members of parliament laughing at us after the last time ... because they had an all-women shortlist, didn’t they, because it was Theresa May and Andrea Leadsom.
“It’s not great if we can’t ever seem to think that the women are good enough. When you’re a woman in politics you’re always the next time – oh it’ll be your turn next time.
“When you’re a woman in politics it’s always, we’re always told we have to pick the best person for the job as if that best person – they never mean the best person, they often mean the best man and that is disappointing.”
Asked if men should sometimes step aside, she said: “If you truly believe in women’s representation, sometimes – and that goes not just for just women in the way that that intersects in other areas – but sometimes passing the mic is the greatest way to show that you truly believe in something.
“But I think it is a very, very difficult argument to say you should stand back if you truly believe you are the best person for the job.”
Ms Phillips also insisted she was not an “uber Remainer” and distanced herself from previous comments that the UK could rejoin the EU.
On Brexit, she said: “People are OK if you are clear and you are honest about what you think and then they will give you a hearing on other issues, so I’m not an uber Remainer.
“I don’t feel the need to represent myself as an uber Remainer, I am a politician who will do what I think is right.”
Five candidates are battling it out to become Labour leader, with Sir Keir and Ms Long-Bailey leading in the polls. Ms Phillips, Lisa Nandy, the MP for Wigan, and Ms Thornberry, the shadow foreign secretary, are also in the running.
Mick Whelan, general secretary of the Aslef trade union, said his union had not decided who it would support – in a blow to Ms Long-Bailey, who will be counting on winning their backing.
He said: “People sometimes forget that the trade unions formed the Labour Party and there used to be a time when they were considered the political wing of the trade union movement. A lot of the movement since we’ve had in 2015 where Ed Miliband did vast work and Jeremy’s driven it further forward – we’d like to see that continue.
“We need to articulate that voice of ordinary people that the Conservative Party seem to have claimed in the short-term because of a one-topic process.”
It comes ahead of the deadline to sign up as a Labour member to vote in the contest on Monday.
Some 14,700 people have signed up as registered supporters already, which allows non-members to vote in exchange for a £25 fee.