Regular warnings about Dame Lowell Goddard’s allegedly offensive behaviour were ignored by Home Office staff and advisers, it is claimed.
The growing concerns were being passed to both Theresa May's special adviser, Liz Sanderson, and to Mark Sedwill, the most senior civil servant at the Home Office, insiders told The Times.
Under the terms of Dame Lowell's appointment in February last year to chair the inquiry, on a £500,000 salary package, only the Home Secretary had the power to sack her.
Criticism is also focused on the explanation given by Amber Rudd, Ms May’s successor in the post, when she was asked to explain Dame Lowell’s resignation in August.
Quizzed by the Home Affairs Select Committee last month, Ms Rudd said Dame Lowell had quit because she was “a long way from home” and “found it too lonely”.
Mr Sedwill was sat next to Ms Rudd as she gave evidence, adding: “That's all the information I have.”
A senior inquiry source told The Times that Mr Sedwill's silence, as the Home Secretary gave her explanation, “smacks of a cover-up”.
The source said: “Mark Sedwill knew the truth. He sat there and said nothing as his boss claimed that Goddard resigned because she was homesick. That wasn't true and he knew it.”
The Times had also revealed that Dame Lowell had been overseas for three months of her first year in office, prompting fierce criticism.
A senior Labour MP who sits on the Home Affairs Select Committee, Chuka Umunna, repeated his call for it to fully investigate the crisis at the inquiry.
He tweeted: “If she agrees to our request to appear b4 @CommonsHomeAffs Judge Goddard wld have the opportunity to respond in full to these allegations.”
The Home Office declined to comment on the allegations about Dame Lowell's conduct, or on what Mr Sedwill and Ms Rudd knew about them.
A spokesman insisted that due diligence checks were made before her appointment, while Ms Sanderson - who now works with Ms May at No.10 - referred questions from The Times to the Home Office.
The newspaper was told by a spokesman for the inquiry that “all human resources issues are confidential”.
There were suggestions that Ms May's failure to act against Dame Lovell was a legacy of the inquiry's earlier troubles, when her first two chairwomen were forced to resign over their perceived links to the establishment.
Questions could also be asked of Professor Alexis Jay, a senior member of the inquiry’s staff, who went on to become its fourth chairwoman.
Dame Lowell has described allegations made in The Times as “falsities”, “malicious” and part of a “vicious campaign”.
The New Zealand high court judge added: “I confirm my absolute rejection of this attack. I am confident that in New Zealand my known reputation from my work over many years will provide its own refutation of these falsities.”
The Times investigation also focused on the close links between the Independent Inquiry into Child Sexual Abuse (IICSA) and the Home Office, despite its purported independence.
John O'Brien, who moved from a senior Home Office role to become inquiry secretary, was recorded as saying that the IICSA was separated from Government by “a low brick wall”.
Of the 167 people working for the inquiry last month, 38 were former Home Affairs staff and another 38 came from other Whitehall departments.