Vieira spent nine years at Arsenal between 1996 and 2005, becoming one of the central figures in the club’s rivalry with United. His battles with Keane became a defining feature of the Premier League during that period.
Speaking to Sky Sports, Vieira confirmed that he had met Ferguson to discuss a move to Old Trafford.
“I had the opportunity to move to Real Madrid a couple of times and I even had an opportunity to move to Manchester United,” Vieira said.
“I had a meeting with Sir Alex Ferguson. Full respect, I couldn't say no to meeting him because he was an important figure in the Premier League.”
Vieira said Ferguson explained his plans for United and outlined a midfield that would have included the Arsenal captain alongside Keane.
“We met and we talked and he explained to me his vision that he had for Manchester United and the team that he wanted to build,” he said.
“He wanted to play me with Roy Keane, (Ryan) Giggs on the left-hand side, David Beckham on the right-hand side and (Paul) Scholes to play behind (Ruud) Van Nistelrooy, so it wasn't a bad team.”
The prospect of playing alongside Keane was particularly striking given the pair’s fierce rivalry as captains of Arsenal and Manchester United.
They had several confrontations during the peak of the rivalry, including a heated exchange in the tunnel before a match at Highbury in February 2005.
Vieira ultimately decided against leaving Arsenal, where he had become a key part of Arsène Wenger’s team.
“But I couldn't leave Arsenal, I was too happy there and we were on the way of building something quite really special for the club and that's the way it was,” he said.
“When you do well, of course there's teams who are interested and then you have to make a decision.”
“But the decision for me was to stay at Arsenal because I felt the love from the club, from the players and from the fans. Everything was there for me to grow up as a player and to achieve what I wanted to.”
Vieira would go on to captain Arsenal to the Premier League title in the 2003-04 season as part of the unbeaten Invincibles side. It was his third league title with the club before he left for Juventus in 2005.
United had been linked with Vieira in the summer of 2003, when he had one year remaining on his Arsenal contract. Arsenal vice-chairman David Dein publicly insisted at the time that the midfielder was not for sale.
Ferguson’s proposed midfield partnership therefore remained a plan that never reached the pitch.