Nick Clegg has admitted that the Liberal Democrats' former chief executive Lord Rennard did not resign from the party purely on health grounds, as the deputy prime minister had insisted at the weekend.

Clegg, who has rejected claims of a cover-up after criticism that he and his party did not do enough when confronted with allegations of sexual harassment against Rennard, had maintained that the peer stood down in 2009 for health reasons. But on Monday the Guardian reported that senior Lib Dems suggested Rennard quit after pressure from Clegg's office.

During his weekly phone-in on LBC radio, Clegg said: "He left on health grounds but of course the issues of his inappropriate behaviour were in the background."

He said that as the party's new leader he had wanted a new chief executive "to represent his values and his priorities. I felt it was time for a change at the top of the professional party." Rennard's ill health was "the immediate reason he stood down", he added.

Rennard also won the influential support of Lady Williams, the veteran Lib Dem peer. She sprang to the defence of both Lord Rennard and Clegg, saying: "Chris Rennard, in my view, is a very fine man, and I think the whole of this has been hopelessly exaggerated.

"Let's be frank, the bad stuff is basically what's happened to Savile. It's abusing children, it's abusing very young innocent women, it's using the temptations of celebrity … and all that, I completely understand that this is terrible, simply terrible. I don't think anything that's been said about our chap is in the same category at all."

Clegg's comments are likely to renew questions about his handling of the issue. He faced criticism after Lib Dem officials spent three days insisting Clegg had not known about complaints over Rennard's behaviour, before he admitted on Sunday that he had been made aware of "indirect and non-specific concerns" in 2008. Rennard denies all accusations.

During the phone-in, the Lib Dem leader was forced to justify his response to the claims despite his attempts to draw a line under the issue before the Eastleigh byelection by launching two investigations.

He denied knowing about the allegations of sexual harassment in 2007. The Telegraph reported that the former MP Sandra Gidley claimed she told Clegg about her concerns after he became leader in December 2007, but this did not imply she necessarily told him in the two weeks remaining of that year.

Clegg said: "I can only tell you the truth as I recollect it now … My recollection is that 2008 is the year the allegations were brought to my attention."

He said specific allegations put by the Telegraph to Jonny Oates, his chief of staff, four days before polling day in 2010 were not passed on to him. He admitted to "very serious mistakes", highlighting problems with the party's channels of communication.

Clegg defended his failure to launch a full investigation in 2008, saying his office was told the women making the claims did not wish to take the matter forward but "just wanted the inappropriate behaviour from Lord Rennard to stop, and that is why he was confronted and a few months later he left".

He said: "You can only launch investigations into allegations when they've been made … That is not something that happened as far as I am concerned until last Thursday [when Channel 4 ran the story]."

He added: "We acted … as we could with what we were told at the time. The women have been let down, there have been some very serious mistakes."

Asked why Oates had not passed allegations from the Telegraph to him, Clegg said: "He felt that four days before a general election, because he knew what the answer was, that I didn't know anything about the allegations, he would just send back an answer straight off to the Daily Telegraph.

"But clearly something went seriously wrong in the organisation as a whole, that people were not talking to each other in the way that they should, and most importantly the people who matter the most are these women who, I can only imagine the anguish, that you've been intimidated and bullied and threatened in the way that they say they were and that you then feel that no one, who should be looking after you, should be looking after your interests, should be protecting your dignity and respect, that people didn't properly listen."

He also had to defend himself from claims that he was seeking to muzzle the press by condemning self-appointed detectives, saying: "I will defend to my last breath [newspapers'] right to say what they want to say."

He praised the way in which Channel 4 had investigated the issue of Rennard's behaviour.