Jose Mourinho last month stripped Pogba of the United vice-captaincy amid reports of frosty relations between the two during the Red Devils' hard start to the 2018-19 season.
"For us, it was the best trophy you can win as a footballer", the Frenchman admitted.
Los Blancos won an unprecedented three-straight Champions League titles during his reign, but fell short of a significant La Liga challenge in his final season. "I'd like to do that".
Ashley Young captained United in Antonio Valencia's absence against Newcastle United last weekend - a game in which Pogba was awarded the Man of the Match accolade.
Pogba told L'Equipe: 'I have never played for France to be captain.
"You do not have to be a captain to speak".
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"You don't have to be captain in order to speak, a leader is not someone who has the armband", he is quoted as saying by The Mirror. It puts pressure on the manager.
While he has been told by United not to speak to the UK media for the foreseeable future, the midfielder has spoken to the French media while he was away on global duty and was asked whether he had an ambition to lead the France team.
Cristiano Ronaldo and Lionel Messi have each won five times in a decade-long stranglehold on the honour with Zinedine Zidane the last French victor after their last World Cup triumph in 1998.
Pogba was the star performer in the second half, setting up a goal and putting in an all round display.
"I've never played in the France team to be captain, being here is already a big thing for me", he said.
"I'm not going to get into it, whether it's (Antoine) Griezmann, Kylian (Mbappe) or Raphael who deserve it much more than me". They gave me confidence as well. "But I hope with all my heart that it's one of the four, because there is [also] NG [N'Golo Kante]".