Labour MP John McDonnell has said he was "sorry" if he caused offence with his remarks about former Tory PM Margaret Thatcher.
At a Labour leadership event on Monday he said that if he could go back in time he would "assassinate Thatcher".
The left-wing MP told BBC Radio 5 live it was meant as a "joke", but he could understand if people took offence.
He also said he would quit the leadership contest if it meant Diane Abbott could get onto the ballot paper.
Campaigning is already under way by the six candidates hoping to succeed Gordon Brown as Labour leader but they must get the backing of 33 Labour MPs by Wednesday to go through to the vote of party and union members in September.
Both the left-wing contenders - Mr McDonnell and Ms Abbott - are well short of the support required although Mr McDonnell has picked up an extra five supporters since Monday and Ms Abbott an extra two.
Mr McDonnell had told the GMB union hustings event that he had been asked what single act he would do to improve the world if he could travel back to the 1980s.
"I said: 'Look, I was on the GMC that Mrs Thatcher abolished, I worked for the NUM and we had the NUM strike, I think I would assassinate Thatcher'." He was applauded by union members in the audience but heavily criticised by commentators.
Nile Gardiner - who worked as an aide to Lady Thatcher after she left office - wrote in his Telegraph blog that Mr McDonnell should apologise "for his disgusting, undignified and menacing words" and attacked them as "a sickening disgrace and a stain on his party".
And Conservative MP Conor Burns told the BBC's Daily Politics Show it was "very distasteful" although he added that Lady Thatcher would find it "absolutely hysterical".
"It was a very silly remark, the man is scrabbling around trying to get the number of nominations to stand. She got more votes in her third general election than her first. She is a towering historical figure who is now an elderly lady and he is trying to appeal to the left to get his nominations in."
Mr McDonnell told the BBC: "I'm sorry if I have caused offence to anyone. It was a joke and in that audience it was taken as a joke... it was taken out of context, I can see if people are upset about that and if I have caused offence to anyone of course I apologise."
He said the audience had taken it as a joke but he understood if people had been offended: "I'm not the sort of person that would want to upset people."
Amid reports that Ms Abbott's supporters had suggested he step aside, he said he would see how the nominations stood at the end of the day and he was still hoping they could both get through.
But he added: "I've also said, if it comes down to it, if it's a choice of me standing and not getting on and me pulling out and getting a woman on the ballot paper that's exactly what I'll do because I think it's important we actually reflect modern society."
Ms Abbott - who was the UK's first black female MP - has complained that the threshold is "ridiculously high" and said if Barack Obama, when he was a senator, had to get the same level of support for his run for the US presidency it "would have been over before it began".
David and Ed Miliband and Ed Balls are already through to the next stage of the contest - Andy Burnham has 26 MPs' support. Ms Abbott has nine backers, Mr McDonnell has 15.
David Miliband has well over the required nominations - 71 - while his brother Ed has 54.
Mr McDonnell criticised them for having "sort of swallowed up nomination after nomination almost as vanity" and not asking their supporters to nominate others to get a "full spread of all the candidates" on the ballot paper.
David Miliband said on Sunday that he only controlled one vote - his own - and he had not yet used it. If another candidate got 32 backers, he would use it to get them on the ballot paper, he told the BBC's Andrew Marr programme.
Mr Burnham said on Sunday he was confident he would get the required 33 MPs by Wednesday.
Lady Thatcher, 84, is expected to visit Prime Minister David Cameron in Downing Street on Tuesday.