The former Chelsea team doctor Eva Carneiro is seeking a public apology from José Mourinho as part of any settlement in her employment case after lawyers failed to reach agreement over six hours of talks in south London.

Carneiro is suing Chelsea for constructive dismissal and has a separate but connected personal legal action against Mourinho, the team’s former manager, for alleged victimisation and discrimination.

The case appears destined for a full employment tribunal pencilled in for 6 June after a private mediation hearing in Croydon – attended by the Chelsea chairman, Bruce Buck, and the influential board member Marina Granovskaia – did not yield a compromise over a financial package for compensation and damages.

The doctor had been criticised by Mourinho and demoted from first-team duties after she and the physio Jon Fearn, waved on by the referee, entered the field of play to treat Eden Hazard in stoppage time at the end of a 2-2 draw with Swansea City on the season’s opening day last August. The Portuguese, manager, who was sacked by the struggling Premier League champions four months later, branded the pair “impulsive and naive”, with their actions having, in effect, left his side, already depleted, down to nine men for the final seconds.

Carneiro left the London South Employment Tribunal, accompanied by her husband, the polar explorer Jason De Carteret, at around 3.30pm on Monday and shook her head briefly when asked if a resolution had been reached. Her barrister, Mary O’Rourke QC, added that proceedings were “confidential” but had indicated earlier that her client would be demanding a full apology from Mourinho as part of any settlement.

“We’re not expecting a resolution today,” she had told ITV News upon arriving at around 9.15am. “The two sides are so far apart financially. And we want Mr Mourinho to make a public apology.”

It is understood the Portuguese, who is keen to return to a managerial role in the Premier League this summer and has been heavily linked with Manchester United, was not represented on Monday, with Chelsea apparently continuing to defend the manager they sacked for a second time – over the team’s poor performances – in December.

Should no agreement be reached out of court before the allocated date in June then a full employment hearing will be held in public. Statements and documents, including emails and text messages, would potentially be made available and the likes of Mourinho, the referee Michael Oliver and other employees of Chelsea, including players and Fearn, who remains at the club, could be called as witnesses.

The club, who filed their defence in the tribunal hearing in December, have consistently declined to comment on internal staffing matters but are keen to avoid having the issue played out in the public eye. That would suggest that there may still be scope for a compromise to be reached over the next three months .

Carneiro has received support from Fifa’s medical chairman, Michel D’Hooghe, who argued she was simply doing her job by seeking to treat Hazard. She did not appear on the Chelsea bench again after the incident, having been dropped from first-team duties, and left the club six weeks later.

Mourinho was cleared of using discriminatory language towards her following an investigation by the Football Association, though the English game’s governing body came in for criticism from Carneiro and its independent board member, Dame Heather Rabbatts, for not interviewing the doctor as part of its own inquiry.