Pogba's comments after the Wolves game spun Mourinho into a fury as he claimed that Pogba thought he was bigger than the club.
"The only truth is that I made the decision that Paul will not be a second (vice) captain anymore but no fallout, no problems at all", Mourinho said after his team were knocked out of the League Cup on penalties by second-tier Derby County.
"The person who made the decision that Paul Pogba is no longer the second captain is the same person that made the decision to make Paul Pogba the second captain. I am a proud manager", Lampard told Sky Sports.
Pogba still turned up at Old Trafford, watching from a private box, but disappeared before the shoot-out.
United are now only seventh in this season's Premier League table.
The only all Premier League tie of the night saw Leicester edge out in-form Wolves 3-1 on penalties after a 0-0 draw thanks to three shootout saves from Danny Ward.
Pogba's future at Manchester United is understood to be in serious doubt. "What a performance from the team and in terms of personality".
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"I've dreamed of this moment, growing up as a City supporter", said Foden.
"Jose was very welcoming as a whole". I know I lost the ball (for their goal).
"I prefer to be proactive, but then we go for it only after the 1-1 and after the red card we are in a little difficulty".
Derby came from behind to lead thanks to a brilliant long-range free-kick from Liverpool loanee Harry Wilson, with Jack Marriott nodding the Rams ahead.
Belgian global Marouane Fellaini netted an equaliser in the fifth minute of added time for 10-man Manchester United after Sergio Romero's sending off. The former England midfielder is only in his second month as a manager after being hired by Derby.
The World Cup-winning France midfielder subsequently questioned Mourinho's tactics in an interview, suggesting they should adopt a more attack-minded approach at home.
"To do it on this stage against this team full of top players with our squad which has some younger boys in it as well - the players were enormous".