Aigner, who confirmed he will retire at the end of the year, is likely to play a key role in the appointment of his replacement, and is known to be close to Lee, who has acted as UEFA's spin doctor in the role of director of communications.
One UK newspaper suggested Lee had pulled out of the race to become chief executive of the English Football Association so he could chase for the UEFA hotseat.
Meanwhile, Aigner has expressed his satisfaction at the state of preparations for next summer's Euro 2004 tournament in Portugal.
As UEFA unveiled the event's official mascot (pictured), Aigner said plans were at 'full speed'.
Said Aigner: "Work on the stadiums is progressing at a good rate and all the other elements needed to guarantee a successful event are gradually being put into place, from the broadcasting of the matches to arrangements for taking care of the supporters and ticket sales.
"On the commercial front, almost all the sponsorship contracts have been signed, the tournament logo was presented last year and the mascot has been unveiled too."
The mascot - Kinas - was unveiled by Dr Gilberto Madail, president of the Portuguese Football Federation and chairman of the board of Euro 2004.
The name Kinas is derived from a significant Portuguese symbol, the five shields in Portugal’s coat of arms.
Said Madail: "Warner Bros. Consumer Products, with their great experience, have succeeded in developing a mascot which embodies the enthusiasm, skill, challenges and fun which come together to make football what it is."
