Jose Mourinho was sacked on Tuesday having overseen a disappointing start to the 2018-19 campaign that sees United in sixth place in the table.
Indeed, this is the Red Devils worst ever start to a Premier League season, with the club currently 19 points behind leaders Liverpool.
The club have announced that, following Mourinho's departure, they will appoint an interim boss until the end of the season, and Goal understands that appointment will be external, rather than one of the club's current coaching staff.
That will give United the time to recruit a permanent replacement ahead of the 2019-20 season.
But, when asked about his potential interest in the job at his pre-match press conference ahead of Wednesday's north London derby against Arsenal in the Carabao Cup, Pochettino insisted he is only focused on Spurs.
He told reporters: "I want to send my best wishes to [Mourinho]. I am so sorry, I know him well, and it's sad news. It's not my business what happens at another club, I just want to send my best wishes to Jose.
"In five years here a lot of rumour has happened, I respect a lot the opinion of everyone. A lot of rumours happen, but it's not my business what has happened. I want to deliver my best show at this football club."
Rio Ferdinand also offered his thoughts on Mourinho's exit, explaining on social media: "I was one of his biggest advocates two and half years ago when he joined and felt he stabilised our club when we needed it.
"You can’t ignore what Jose has achieved as a manager previously but I feel the time has come for fresh ideas at Man United."