Martinez met with manager Michael Carrick and club executives after training on Thursday, two days after saying the club needed greater investment to help the team compete at the highest level following their 2-0 loss to the Spanish side.
"Today I met with the club's leadership and the manager," Martinez said in an Instagram post, in his second statement since the interview.
"It was a very positive conversation where we were able to exchange views and speak openly about what happened after Tuesday's game."
A club source said any internal disciplinary action would remain private, adding it had been Martinez's decision to apologise publicly and all parties considered the matter closed.
Martinez acknowledged his frustration following the defeat had got the better of him and apologised for choosing the wrong moment to air his feelings.
"Everyone knows that I am a passionate person and that I live football with great intensity," he said.
"The frustration of the defeat led me to express how I felt at a time and place that perhaps were not the right ones, and for that I want to apologise."
The Argentina World Cup finalist had singled out director of football Jason Wilcox in the post-game interview, saying United needed more signings to compete with their rivals and build a squad capable of challenging in Europe.
Martinez said his comments had not reflected the unity required at Old Trafford and stressed his belief that everyone at the club was working towards the same goal.
"Our club is at its best when we are all united, and I recognise that my words in that moment did not come up with that," he said.
"I know that every single person at Manchester United is doing everything possible to achieve our shared ambition, bringing the success that this club and its supporters deserve."
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