Alison Smith, a former councillor in Aberdeen, who made complaints about the party's former chief executive tweeted that Mr Carmichael "suppressed concerns" over his colleague's behaviour in his past role as chief whip.
The Orkney and Shetland MP replied: "I did not suppress or cover up anything" and said he had met one of the female complainants but found he could not see an obvious way of resolving the issue.
"She wanted to see action taken against Chris Rennard without making a public statement herself. I left it saying I would speak to her again but for a variety of reasons I didn't," he said.
Ms Smith, a politics lecturer at Oxford University, later removed her tweet about Mr Carmichael.
Mr Carmichael had described the Rennard row as "unfortunate", admitting it was not good for any party "to be seen to be washing its dirty linen in public".
Mr Carmichael called for an independent mediator to be swiftly brought in to avoid what one party source described as "mutually assured destruction".
He added: "We need to deal with it quickly and cleanly now, bringing in somebody who would be trusted by both parties, to sit them down and say - if you are motivated by the best interests of the party, we must find a way through this."
Friends of Lord Rennard - whom one party source said had been "destroyed" by the row - made clear the party's former election strategist would be willing to avoid legal action if his suspension was lifted first.