West Ham boss Allardyce, who is expected to leave Upton Park at the end of this season, has enjoyed a lengthy managerial career, taking charge of the likes of Bolton Wanderers, Newcastle United and Blackburn Rovers before pitching up in east London in 2011.
But the 60-year-old said being the boss at Old Trafford represents his "dream".
In an interview with BT Sport, Allardyce revealed: "If I didn't say West Ham were the dream club to manage then I'd be in trouble with West Ham fans, but there has always been one dream club to manage for me - and that would be Man United.
"Playing at Old Trafford myself and then going back there as a manager - I think probably in this country, they are the one."
Allardyce went on to explain that his "biggest disappointment" remains losing out to Steve McClaren in the race to replace Sven-Goran Eriksson as England manager in 2006.
"I had done two interviews for the job and [then Football Association chairman] Brian Barwick called me to say it was between me and Steve. He said 'we will let you know'," Allardyce added.
"I never got a phone call."
