Pogba's future has been the subject of much speculation in recent years, amid persistent questions over his form and fitness at Old Trafford.
United forked out £89 million ($115m) to re-sign the midfielder from Juventus in 2016, but he has since struggled to live up to that price tag.
The 27-year-old has often been made a scapegoat following defeats or poor team performances, and he appeared to have reached a breaking point at the end of the 2018-19 campaign.
Pogba openly admitted to being ready for a new challenge away from Manchester in the summer of 2019, with Madrid mooted as a possible next destination, but Ole Gunnar Solskjaer ultimately managed to convince him to stay put for another year.
A serious ankle injury significantly limited the Frenchman's contribution last term, and it wasn't until football returned after lockdown in June that he rediscovered his best form.
The World Cup winner ended up playing a key role in United's run to a third-place finish in the top flight, but has found himself under intense scrutiny once again at the start of the new campaign.
The Red Devils have only picked up three points from a possible nine in the Premier League, with the international break coming as a welcome reprieve for Pogba amid mounting criticism.
It has been reported that Madrid boss Zinedine Zidane remains a keen admirer of the playmaker, and he opened the door to a potential transfer at a press conference following France's 7-1 friendly win over Ukraine in midweek.
Quizzed on his fellow countryman's rumoured interest in him, Pogba told reporters on Thursday: "We have all heard that things have been said. What to say? Yes, all footballers would love to play for Real Madrid. It might be a dream.
"It is a dream for me, why not one day? But I play for Manchester United and I love my club. I perform for Manchester United, I have fun and I want to do everything to put the club where it deserves to be .
"I will give my maximum, like my team-mates."
Pogba, who has just entered the final year of his existing contract at Old Trafford, went on to reveal that he has yet to be offered a renewal, although United do hold the option to extend the deal by 12 months.
"I have not spoken with Ed Woodward (executive vice-chairman of Manchester United)," he said. "We did not talk about a renewal. For now, I am in Manchester and I mainly think about getting back to the best of my form.
"I think there will be a moment when the club will come and talk to me and maybe offer me something, or not. So far, nothing has happened. I can't tell you something that I don't know."
United suffered a humiliating 6-1 loss at home to Tottenham on Sunday, which increased the pressure on Solskjaer as he bids to bring the club's three-year trophy drought to an end.
Pogba says the whole squad are still pulling in the same direction despite another damaging setback, as he added: "We know it's not easy, we've had difficult times. It's difficult at the moment.
"We are fighting, that's what it takes. We know that we will go through low times as a footballer.
"You have to have a strong mental strength to come back to the best level and to find the place which suits us to put the club where it deserves to be. It deserves to be at the top.
"To win titles and the championship, these are the ambitions of the club. We know that it is not easy, but that remains our objective. We are fighting for that."