As well as United and Chelsea, Robben is also believed to interest clubs in Italy and Spain.
But his father Hans, who acts as his agent, said recently that if his son was to leave Eindhoven it would only be for United. He said: "It's either Manchester United or PSV."
Robben, rated one of Holland's best young prospects, is contracted to PSV until 2007.
The competition between Chelsea and the champions has grown since Abramovich's arrival. The signing of Argentine star Juan Veron from Old Trafford was overshadowed by Chelsea's dramatic swoop to lure away United's chief executive Peter Kenyon.
The clubs were again in competition earlier this month when Chelsea inquired about Fulham striker Louis Saha. United eventually won the battle for Saha, who was in Manchester this week to finalise a £12million move. The loss of Robben will be a bitter blow to Ferguson, though.
Meanwhile, Charlton chairman Martin Simons has told Chelsea's billionaire owner Roman Abramovich to keep his money and "get lost".
Chelsea have had two offers for Scott Parker rejected - the latest worth £7.5m plus bonuses based on appearances.
But Simons doesn't want to sell. He said: "Chelsea must be shown they can't take our players without us putting up a fight.
"We are fourth in the league, they are third. In the last few weeks we have been catching them up. So it hurts when a club with their buying power see the club beneath them and say, 'Oh, let's spend a few million and take their best player'.
"Well I say, 'Get lost Abramovich', and I think I speak for the rest of football. Somebody has to make a stand.
"We want to send out a message that our best players are not for sale." However, Simons admitted every player has his price. He said: "Obviously, money often talks. If they were to offer, say, £25m then that is an offer you can't refuse."