Football Association chairman Greg Dyke has told officials that Chelsea manager Jose Mourinho committed "a failure of his personal judgement and public behaviour" in his criticism of team doctor Eva Carneiro, it can be revealed.
In a letter to FA council members, a copy of which has been obtained by Press Association Sport, Dyke says Mourinho should have apologised.
The letter was sent following Wednesday's FA board meeting, since when independent board member Heather Rabbatts has launched a stinging attack on the FA for its "seriously disappointing" handling of the Carneiro case.
Rabbatts, head of the FA's inclusion advisory board, said she had "major concerns" regarding the FA's disciplinary process which saw Mourinho cleared.
Dyke's letter makes clear his misgivings over Mourinho's behaviour. Carneiro was dropped from first-team duties after an incident on the opening day of the season when she was criticised by Mourinho for going on to the pitch to treat Eden Hazard. She has since parted company with Chelsea.
Dyke says in the letter: "There have been some well-documented issues of late around equality and inclusion in the game, an issue where it is vital we continue to show clear leadership.
"I felt the handling of the case of the Chelsea doctor, Eva Carneiro, was a good example of this. We supported Heather Rabbatts' strong statement on the matter earlier in the month.
"Personally I don't think Mr Mourinho comes well out of the whole saga - he clearly made a mistake in the heat of a game, and should have said so and apologised.
"Instead he has said very little and Miss Carneiro has lost her job.
"Our regulatory team have investigated this and whilst Mr Mourinho has breached no rules it was clearly a failure of his personal judgement and public behaviour. This should be seen as such by the game."
Mourinho was cleared by the FA on Wednesday of making discriminatory comments towards Carneiro. It has emerged that the FA did not ask to interview Carneiro, although it is understood it asked her lawyers if they wished to provide any evidence.
Rabbatts told Press Association Sport: "The FA's reaction to the treatment of Dr Eva Carneiro has been seriously disappointing. I have major concerns over the way in which the disciplinary process has been conducted and the lack of an organisational response to the wider issues raised by this case.
"We had an announcement late yesterday relating to a high-profile incident which occurred on August 8 and yet it would appear that during that time no witnesses were requested to speak to the FA, including Dr Carneiro, and in the course of the investigation some media were reporting it was likely that no charge was to be brought."
Rabbatts added: "This is on top of a previous case when clear evidence of sexist and abusive chanting from groups of supporters against Dr Carneiro was apparently not seen as sufficient for a charge to be raised.
"This is not only a personal tragedy but is a setback for player welfare and sends a terrible message to other medics and all those girls and women who aspire to play a role at the top level of professional football."
The Football Medical Association (FMA), the body which has been liaising with Carneiro, said expert opinion on the words used by Mourinho was divided.
FMA chief executive Eamonn Salmon said: "As expert opinion would appear to be divided regarding the video evidence and content matter, the FMA is surprised that Dr Carneiro was not interviewed as part of the investigative process."
The FA said on Wednesday an independent academic expert in Portuguese linguistics had analysed footage of the incident, and announced it ''is satisfied that the words used do not constitute discriminatory language under FA rules''.