Relegation-threatened Livorno have sacked Domenico Di Carlo after just 88 days in charge and re-appointed Davide Nicola.
Virgil van Dijk claims he knows nothing of interest from English clubs and is keen to prove himself in the Champions League with Celtic.
Paolo Di Canio's exit from Swindon, and West Ham's dip in form, had led to speculation that Gold and his partner David Sullivan were thinking of bringing the Hammers icon in to replace Allardyce at the end of the season.
Gold retweeted posts from two West Ham fans on Tuesday calling for Di Canio to replace Allardyce, as well as two more of the opposing view.
Gold insists the club will not enter talks with Allardyce about extending his contract until the team are safe from relegation, but he did throw his weight behind the former Blackburn and Newcastle manager on Tuesday evening with a tweet of his own, which read: "Always interesting to see the different opinions from people on twitter. My view is clear though - Big Sam is our man! dg."
Allardyce brought West Ham up from the npower Championship in his first season at Upton Park, but his direct style of play has alienated a section of the Hammers faithful.
Allardyce's men sit seven points above the Barclays Premier League relegation zone with 12 games of the season remaining.