Bundesliga side Dortmund completed a deal to sign Witsel from Chinese side Tianjin Quanjian earlier this week.
The 29-year-old midfielder, who impressed for Belgium in their run to the semi-finals of the World Cup, signed a four-year deal with the German side in a deal worth £18m.
However, the player's agent, Paul Stefani, says he advised his client to ignore an approach from United after the Premier League side signaled their interest in a deal.
"We chatted for a moment, Axel says: 'Maybe I want to leave China'. I say to him laughing: 'Axel, I promise that I will come up with something really great!'", Stefani told Belgian outlet HLN.
"I remembered that a few months earlier I had been to Dortmund, where I met someone from the club. I call that man and he brings me into contact with the management.
"Sports director Michael Zorc and CEO Hans-Joachim Watzke invited me to come and talk at a time when Axel was at the World Cup in Russia. There I heard that Dortmund were looking for a player like Witsel.
"I explained to them that the Portuguese lawyer of Witsel has negotiated a buy-out clause of €20m with Tianjin. That sounded like music to them, because they thought that Tianjin would ask for an amount of around €60m for Axel. When the transfer later started rolling, the Chinese still tried to get out of that buyout clause, but it was conclusive.