The 29-year-old joined Dortmund earlier this week from Chinese club Tianjin Quanjian in an £18m move, but was sought after by many of Europe's top clubs after a successful summer World Cup campaign with Belgium.
And the player's agent, Paul Stefani, told Dutch-Belgian news outlet Het Laaste Nieuws that Witsel was a transfer target for Red Devils boss Jose Mourinho - on a shortlist which also included Chelsea midfielder N'Golo Kanté.
Stefani said, "There were other candidates. Manchester United wanted him, Napoli too. But I told Axel: 'At Dortmund you are the number one, at Manchester you are one of many...and at Napoli too.'
"I know from a very good source that Mourinho said he wanted Witsel, but not in the first place. N'Golo Kanté topped his list. Eventually Axel said: 'Do me a favour and finish it as quickly as possible, because I really want to go to Dortmund."
Missing out on Witsel is yet another blow for United boss Mourinho, who has been frequently frustrated by his side's lack of new signings during this summer transfer window.
He confirmed this frustration to reporters after his side's 4-1 International Champions Cup defeat to Liverpool last month, saying that he had given chief executive Ed Woodward a long list of players to sign - but there have been no new arrivals since.
Mourinho said, "I would like to have two more players, I think I am not going to have two. I think it's possible I am going to have one.
"I gave a list to my club with five names a few months ago and I wait to see if it's possible to have one of these players."
The summer transfer window slams shut for Mourinho's United and the other English clubs on Thursday afternoon at 17:00 BST.