The Netherlands midfielder joined United in a deal that could eventually be worth £39.1million (€44m), signing a five-year contract with the option of a sixth.
Van de Beek had spent his entire career at boyhood club Ajax, but is relishing linking up with his new team-mates at Old Trafford after being given such a warm welcome.
He posted on his Twitter page on Thursday: "Very excited to be the newest edition [sic] of the #MUFC family.
"Your enthusiastic responses to my announcement make me feel at home already! Can't wait to join the team in Manchester.
"But first... focus on the Dutch Squad in preparation for the upcoming Nations League matches."
Swart said the coronavirus pandemic impacted the market but told Ajax Showtime: "Manchester United is certainly the right choice.
"They used to be a championship-winning club and they now want to build that way again."