Maguire is starting his seventh season at the club following his £80 million ($107.7m) move from Leicester City in 2019.
In January, United triggered an option to extend his contract until 2026.
But the centre-back has said that did not stop clubs from expressing their interest during the summer transfer window.
"I think there were a couple of clubs who maybe enquired or spoke with them [United] and I think they got a quick response," Maguire said.
"I'm pretty sure the club have made it aware this summer that I can't leave on any terms with other clubs enquiring about things and my position with my contract.
"We're in a good place, positive as a club and I feel like the hierarchy has come in and Jason [Wilcox] and the manager, I feel like they're taking it in the right direction.
"I think it's been, since I started six years ago to where it is now, it's in a completely different place in terms of the structure behind the management staff."
Maguire would have been out of contract this summer had United not taken up their option to extend his deal by an additional 12 months.
Now into the final year, he can begin talking to clubs outside the Premier League from Jan. 1 if there is no new agreement with United.
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The 32-year-old is not giving much away about his future, but suggested his preference is to stay at Old Trafford.
"Last year, the clause was in their hands so there was no option for me there. There was no talking," Maguire, who was speaking at a Manchester United Foundation's holiday camp at Stretford High School, said.
"It was just they activated it and it got extended. This year, obviously I'm up at the end of the year.
"I'm sure over the next few months they'll sit down and we'll have to have a conversation about where we want to go and if they want to extend or obviously the transfer window will open again in January.
"I have something in my mind about what I want to do and what I want to be. I don't want to put it out there to everybody but it's an amazing club to play for and you'd be silly if you wanted to jump out of it as soon as you could."
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