Angela Rayner says she would be “happy to sit down” with the PM after she faced criticism for calling him “scum”.
Labour’s deputy leader has stood by the comment she made during a fringe event at the party’s conference.
Taking to Twitter on Monday, she repeated her accusations that Boris Johnson had made “racist, homophobic and sexist” remarks.
And she posted pictures of headlines where the PM had refused to say sorry for his own controversial comments.
Ms Rayner said: “If he withdraws his comments and apologises, I’ll be very happy to apologise to him.”
But Labour’s Shadow Scotland Secretary Ian Murray told the New Statesman she should apologise as politicians should “lead by example”.
“I think it would be worth an apology, and I think it would do an awful lot of good to Angela to apologise as well – and I think it would do an awful lot for our political discourse.”
Shadow Trade Secretary Emily Thornberry said she would “probably apologise” if she had used the word “scum” – but added that Ms Rayner had the “evidence” to back up her other accusations.
Conservative Party co-chair Oliver Dowden said: “Voters have heard the message loud and clear from Labour conference: if they support Conservative values or this government they are ‘scum’.
“Instead of looking down our noses at voters and throwing around playground insults, we are focused on bringing people together and delivering for them.”
The row began on Saturday night when Ms Rayner was reported to have called Tory ministers “a bunch of scum” at a conference event.
On Sunday, she stood by her comments, telling the BBC: “I will apologise when Boris apologises for saying the comments he has made, I will retract that he is scum.”
Labour leader Sir Keir Starmer said it was “not the language I would have used” but did not call for his deputy to apologise.
Ms Rayner renewed her attack on the PM on Monday, tweeting four headlines about Mr Johnson refusing to apologise over remarks he made about Muslim women and gay men.
She added: “Boris Johnson also called the children of single mothers ‘ignorant and illegitimate’.
“According to Boris Johnson, when I was a young single mum I should have been pushed into ‘destitution on a Victorian scale’.
“So you can apologise for those comments as well, prime minister.”
‘Heat of the moment’
Asked about the comments on BBC Two’s Politics Live, shadow trade secretary Emily Thornberry said her shadow cabinet colleague’s initial comments had been made at a fringe meeting “where there may have been drink partaken [and] where it is noisy, rowdy”.
She added: [Ms Rayner] was putting forward evidence of him being racist, misogynistic and then, in the heat of all that, she calls them ‘scum’.
“She probably shouldn’t have said ‘scum’ but the rest of it of course, there is evidence of it.”
Ms Thornberry said it happened “in the heat of the moment”, before adding: “I personally would apologise if I had [used the word scum] but I understand the context of it, I can see how it has happened.
“I don’t think this is the biggest thing that has happened at party conference.”