Mourinho has been resting at home in Setubal since leaving Chelsea by mutual consent in September.
He has given no indication of his next career move.
McClaren replaced Sven-Goran Eriksson after last year's World Cup finals in Germany but his reign was doomed after England lost 3-2 to Croatia at Wembley and failed to qualify for Euro 2008 in Austria and Switzerland.
FA chief executive Brian Barwick and head of soccer development Trevor Brooking have been charged with finding a new manager and have been canvassing opinion from leading figures in the game.
Mourinho had been one of the three favourites, along with Italian frontrunner Fabio Capello and his compatriot Marcello Lippi, who won the 2006 World Cup with Italy.
Capello has expressed an interest in the job while Lippi told BBC sport on Monday he had had "no contact at all" with the FA.
Other names in the frame include former Germany striker and World Cup coach Juergen Klinsmann, Aston Villa manager Martin O'Neill and former England forward Alan Shearer.
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