The east Londoners sit 19th in the Premier League and suffered an embarrassing 5-0 defeat at the hands of Championship outfit Nottingham Forest in the third round of the FA Cup on Sunday.
In an open letter posted on West Ham's official website, Sullivan and Gold acknowledged the "seriousness of our current predicament" but urged supporters to stick behind Allardyce and his team.
"We know Sam has not lost his ambition or desire and is committed to making West Ham a great Premier League club," read the letter.
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Allardyce has been hamstrung this season thanks to the long-term injury to striker Andy Carroll, as well as a host of problems throughout his squad, a situation acknowledged by the co-chairmen.
They added: "Our record signing, Andy Carroll, continues to make good progress following his return to first-team training and he should be back on the pitch very soon.
"We are confident that the return of all of our injured players will lead to an upturn in our fortunes."
There was more positive news on the transfer front, though, where West Ham are expected to bring in Lacina Traore on loan from Monaco, as revealed by Goal on Saturday, while a loan move for Sheffield Wednesday centre-back Roger Johnson is expected to be confirmed imminently.
Gold and Sullivan added: "We are pleased to confirm we are close to securing one of our key striking targets. A defender is high on the list of priorities as well and progress is being made on that front."