The Red Devils submitted a bid to RB Leipzig earlier this week, with Sesko preferring the move to Old Trafford rather than to Newcastle United, who were also keen on his services.
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In an interview with the club's official website, the 22-year-old said: “The history of Manchester United is obviously very special but what really excites me is the future.
“When we discussed the project, it was clear that everything is in place for this team to continue to grow and compete for the biggest trophies again soon.
“From the moment that I arrived, I could feel the positive energy and family environment that the club has created. It is clearly the perfect place to reach my maximum level and fulfil all of my ambitions."