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It has been alleged that Mr Johnson advocated for a role for the woman in 2008, just weeks after they met.
But the appointment is said to have been blocked because his colleague, Kit Malthouse, now a minister in the outgoing PM’s caretaker Cabinet, said the pair appeared to have an inappropriately close relationship.
The Sunday Times reported that the woman was upset by what happened, confronting Mr Johnson about it several years later.
Daisy Cooper, the Lib Dems’ deputy leader, called for an investigation into the matter.
She said: “The woman’s account is deeply distressing to read and it’s clear this must be investigated immediately, either by a parliamentary body or City Hall authorities.
“British politics has been repeatedly dragged through the mud in recent weeks.
“It is utterly depressing that not only has another politician in high office been accused of abusing their power, but that it is now the sitting Prime Minister who faces serious questions.”
According to the Sunday Times report, the woman secured an interview for the role, but felt ill-qualified, and was unsuccessful.
The newspaper said she confronted Mr Johnson nearly a decade later, during the #MeToo movement in 2017, when he admitted pushing her forward for the post.
These awful allegations add to the lies, the law breaking and the security breaches - this man is not fit for office and should go now
He is said to have repeatedly apologised, telling the woman: “I’m very, very sorry, your unhappiness about this.”
But he denied any wrongdoing, according to the newspaper, saying: “I don’t believe that I was malicious and I don’t believe that I would have been aggressive or anything like that. I think that would be not fair.”
Labour’s deputy leader, Angela Rayner, said: “These awful allegations add to the lies, the law breaking and the security breaches – this man is not fit for office and should go now.”
A Downing Street spokesman said: “This not about his time as PM and no public interest as I see it. And we don’t talk about his private life.”