Two members of Theresa May’s panel inquiring into child sex abuse are facing calls to resign after being accused of sending threatening or insulting emails to victims who had criticised the inquiry.

Lawyers for one abuse survivor have written to the home secretary to complain of a string ofunsolicited communications, including an allegedly threatening email sent two days before an official meeting that both panellists and an abuse survivor were due to attend.

The victim, who is on medication for post-traumatic stress disorder, was left too anxious to attend the “listening meeting” in November. The development will be a huge embarrassment to May, who has already seen two chairs to the inquiry stand down since it was launched in July, both over conflicts of interest. A source close to the inquiry’s secretariat said the emails should not have been sent, leaving the fate of the panel members in doubt.

The inquiry has been set up to consider whether, and the extent to which, public bodies and other non-state institutions have taken seriously their duty of care to protect children from sexual abuse in England and Wales from 1970 to the present day.

At a meeting on Friday between government officials and victims of abuse, survivors were told by the Home Office’s director of safeguarding that the allegations against panel members were being examined by an unnamed QC. Survivors told officials they had “lost all confidence” in the inquiry as it was currently constituted.

Andrew Lavery, whose whistleblowing has led to a police investigation into abuse by monks at a Catholic boarding school in Scotland, has complained that he was left highly distressed after receiving an email from panel member Graham Wilmer.

A letter sent by Lavery’s solicitors at Leigh Day claims that their client, who first had contact with Wilmer in 2013 through his charity, the Lantern Project, which offers help to abuse victims, “has received a number of unsolicited communications from Mr Wilmer following the end of their telephone contact”.

The letter adds: “These have included repeated requests from Mr Wilmer asking Mr Lavery to get in contact with him. Mr Lavery has not responded to these requests. Mr Lavery’s key concern is raised in relation to an email which he received from Mr Wilmer on 5 November 2014. This was sent following Mr Lavery’s conversation on 5 November 2014 with a member of the child sex abuse inquiry secretariat about attending a meeting with the inquiry on 6 November. Mr Lavery regards the contents of the email to be threatening. Mr Lavery is of the belief that as a panel member, Mr Wilmer should not be contacting individual survivors in this manner. He was very distressed by the contents of the email and consequently did not feel comfortable about attending the meeting.”

Wilmer, who was himself abused as a child, has denied that he intended the email to be threatening. His email, which has been seen by the Observer, suggests he was provoked into emailing Lavery by a social media row between a separate abuse victim and his adult son, Rory, who had sought to confront those who criticised his father’s place on the panel.

In his email, which accuses his critics of “crossing the line”, he writes: “I’m looking forward to meeting you on Friday. I’ve watched and listened to your media interviews over the past few days, and I’ve looked at all the Twitter feeds you and Ian [McFadyen] have pumped out. The insults you have issued about me, I have no problem with; you are entitled to your opinion. However, what has been said about my son, Rory, is a different matter.”

Catriona Rubens, a lawyer for Leigh Day, told the Observer: “The panellist’s private communications with our client, a man who is deeply traumatised by his experience of abuse, certainly seem ill-advised to us.”

Lavery, who lives in Newcastle, said that he did not want Wilmer to be humiliated, but that he needed to stand down for the sake of the credibility of the inquiry.

Meanwhile, Peter McKelvie, a former child protection manager whose allegations led to the launch of the ongoing Operation Fernbridge police inquiry in 2012, has complained about an inappropriate email from a second panellist.

After McKelvie publicly raised questions about Barbara Hearn’s appointment, she falsely accused him of posing as a female blogger who had been causing her distress.

McKelvie, 65, who has no presence on social media, said he would continue to call for Hearn’s resignation because of her previous employment at the National Children’s Bureau, where a leading member of the Paedophile Information Exchange, Peter Righton, worked as a consultant between 1972 and 1974.

McKelvie also highlighted a potential conflict of interest in Hearn’s connections with social work managers from the Labour-controlled Islington council in north London, when there was widespread sexual abuse in children’s homes. Hearn was appointed to the NCB in the 1990s by John Rea Price, who had been director of social services at Islington council.