Di Canio, who continues to be a fans' favourite at Upton Park, ended a tumultuous relationship with Swindon earlier in the week when he resigned after claiming "broken promises" had made his position untenable.
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Gold retweeted posts from fans calling for Di Canio to replace Allardyce but then reaffirmed his own position by tweeting: "Always interesting to see the different opinions from people on Twitter. My view is clear though - Big Sam is our man! dg."
West Ham have enjoyed a relatively positive return to the Premier league after being promoted from the Championship under Allardyce last season.
They currently sit 11th in the table but are in the midst of a poor run of form that has seen them go out of the FA Cup to Manchester United and win only once in their last six outings.
An official statement from the club regarding their manager's position read: "We will sit down and discuss Sam's contract situation once our Premier League status has been confirmed and not before.
"This is purely because the three of us [Allardyce, Gold and co-chairman David Sullivan] don't want anything to distract attention from our aim of finishing as high up the league table as possible."