The 22-year-old centre‑back arrived at Manchester United’s Aon Training Complex on Wednesday morning after the two clubs agreed a fee understood to be €35m (£31m).
José Mourinho, the United manager, told the club’s official website: “Victor is a very talented young player, who has a great future ahead of him at United.
“Our season last year showed us that we need options and quality to add depth to the squad and Victor is the first to join us this summer. I know that our fantastic group of players will welcome him as one of us.”
Lindelof had arrived in Manchester fresh from international duty with Sweden, where he was an unused substitute in the teams’s 1-1 draw with Norway on Tuesday night.
He has been a long-time target for United as they look to bolster their defensive options for the club’s return to the Champions League next season.