Asked about taking the job, Wenger, 66, said: “One day, if I’m free, why not?” Pochettino, 44, said: “Yes, why not? In the future.”
Both said they are committed to their current clubs.
Allardyce, who replaced Roy Hodgson following England’s disappointing Euro 2016, left by mutual agreement on Tuesday after the Daily Telegraph claimed he offered advice on how to “get around” rules on player transfers.
England Under-21 boss Gareth Southgate will take charge of the senior team for the next four matches, including three World Cup qualifiers, starting with Malta at Wembley (8 October), Slovenia away (11 October), Scotland at home (11 November) and Spain in a friendly (15 November).
Southgate could be an option for the permanent role depending on his success, while Bournemouth boss Eddie Howe, Crystal Palace manager Alan Pardew and ex-Hull City manager Steve Bruce have also been mentioned as candidates.
Frenchman Wenger is the longest-serving – and most successful – overseas manager in England, having won 15 trophies during his 20-year tenure at Arsenal, including three Premier League titles and six FA Cups.
Arsenal, who are unbeaten in eight games in all competitions this season, play Burnley on Sunday at 16:30 BST aiming for a fifth consecutive Premier League victory.