Three of the four women who accused former Lib Dem chief executive Lord Rennard of harassment have called for the peer to be "kicked out" of the party.
Bridget Harris, a Lib Dem activist, has welcomed Lord Rennard's apology, but said his position within the Liberal Democrats was untenable.
"The real question here is, 'Should the Lib Dems accept him back on the benches in the House of Lords?' And I don't think the answer to that is yes", Ms Harris told BBC News.
Another activist who made the claims, Susan Gaszczak, said, "We really have got to draw a line under this. The only way to do that is actually to say, 'Lord Rennard, it's time to go.'"
A third woman, former Welsh Assembly candidate Alison Goldsworthy, told Channel 4 News the apology was a start, but stressed she thought Lord Rennard should be expelled from the Lib Dems.
One of the four female Lib Dem activists who accused Lord Rennard of harassment has said she has mixed feelings after receiving a written apology from the peer.
Former Welsh assembly candidate Alison Goldsworthy told Channel 4 News:
I am relieved that some acceptance has now started to come forward from Chris Rennard that what he did was not okay but very frustrated that I have had to spend the last 15 months talking about things that are really very private and quite intimate to me to be able to achieve it.
If he had accepted and offered this apology up 15 months ago then things may be very different.
Ms Goldsworthy said "the fact that Chris Rennard has now accepted that his behaviour was untoward" meant Lib Dem leader Nick Clegg should expel him from the party.
The Liberal Democrats have confirmed appeals from both sides are ongoing following the investigation into the harassment claims against Lord Rennard.
A statement issued in response to reports of the peer's written apology said:
An appeal has been submitted by three of the original complainants against the outcome of Alistair Webster's investigation.
Separately, an appeal has been submitted by Lord Rennard against the disciplinary process for bringing the party into disrepute by his failure to apologise, and his conduct after the publication of Webster's conclusions.
Both appeals will be considered by the English Appeals Panel in accordance with the rules of the party. We cannot comment further on any aspect of that process while it is ongoing.
Former Liberal Democrat chief executive Lord Rennard has issued a written apology to four party activists who accused him of harassing them.
The peer said "he may well have encroached upon personal space" and added he would like to "apologise sincerely for any such intrusion," which he said "would have been inadvertent".
Lord Rennard wrote that he was issuing the apology after receiving the report by Alistair Webster QC, who conducted an independent investigation into the allegations and the party's handling of them.
Mr Webster found no evidence to act against Lord Rennard but recommended he consider apologising.
A statement issued by his lawyer said: "Lord Rennard wishes to achieve closure of the contentious issues that have been in the public domain for the last 14 months by expressing his apologies to Susan Gaszczak, Alison Goldsworthy, Bridget Harris and Alison Smith."
Liberal Democrat peer Lord Rennard has given his party until Thursday to lift his suspension or face legal action at the High Court in a legal letter issued at the party's headquarters.
Former Lib Dem chief executive was suspended on January 20 after failing to apologise to a number of women who had made sexual harassment complaints against him despite Nick Clegg telling him: "No apology, no whip".
Rennard's letter is understood to name individuals involved in the suspension decision, including the chairman of the Lib Dem regional parties committee Mike Wheatley and the chairman of the English Council of Lib Dems Peter Ellis.
A party spokesman said: "We have received a legal letter last Thursday. It does not constitute serving legal papers."
Miriam Clegg has described claims in the Daily Telegraph that she influenced her husband's response to the row with Lord Rennard as a "complete fabrication".
The newspaper claimed the 45-year-old was "furious" at Lord Rennard's refusal to apologise to the woman who accused him of sexual harassment, and had "shaped" Nick Clegg's response to the row.
Earlier, Nick Clegg angrily denied suggestions his wife was responsible for his tough stance on the issue, saying it was "just not on" for the media to involve his family in the dispute.
"It's total garbage, it is a complete work of fiction. Look, you can make up stuff about me, as indeed people do day in day out, have a go at me, but just don't drag Miriam into it, don't drag politicians' families into it. That's just not on," the Liberal Democrat leader said.
Nick Clegg has insisted he will not back down over his demands for Lord Rennard to apologise to four women who made allegations about the former chief executive's behaviour.
Lord Rennard has threatened to take legal action after he was suspended for refusing to apologise, sparking fears that the party could be torn apart by the row.
Speaking during his weekly phone-in on LBC radio, the Liberal Democrat leader stressed that the conclusions of an independent report into the situation had to be enforced.
"Just imagine if Alistair Webster (QC) had recommended that an apology should be issued and I came out and said 'Thank you very much, we will file away the report, and by the way, none of the recommendations need to be accepted or adhered to'. There would have quite rightly been outrage."
Mr Clegg said Lord Rennard was not acting in his own best interests in refusing to say sorry, arguing it was the "human thing" to do even where distress had been caused "inadvertently".
Alison Smith, one of the women who made allegations against Lord Rennard, has said she would be happy for a report carried out by Alistair Webster QC into the claims to be published.
She told BBC Radio 4's Today programme: "As far as I understood, Lord Rennard has seen what I said. I haven't seen the final report either and I would like to as much as the next person."
Asked if it should be published, she replied: "Yes, actually I do, but actually I've appealed on the grounds that too much emphasis was placed on intent. I think the crux of the matter was whether he brought the party into disrepute, not whether he intended to behave as he did."
Lord Rennard has denied the allegations and has said he will not apologise to the women who complained about him.
Former Liberal leader Lord Steel has told ITV News the party is in an "awful mess" over the Rennard row and it needs "to get a grip" of it.Read the full story ›
A senior Lib Dem source has said that "obviously we disagree" with Lord Steel and that reinstating Lord Rennard "is not going to sort it out".
The former Liberal leader had told ITV News the party should lift the suspension on the former chief executive, who is accused of bringing the party into disrepute by refusing to apologise to women who complained about him.