The London club are also second from bottom in the Premier League, having won just one of their last 13 games, and face the imposing task of a Capital One Cup semi-final first leg at Manchester City on Wednesday.
Normally a vote of confidence from a board spells the imminent departure of a manager but West Ham owners David Gold and David Sullivan appear to have absolved Allardyce of any blame for their plight.
The pair said the manager could hardly be expected to cope in the third round at Forest on Sunday with so many players out injured.
"We know Sam has not lost his ambition or desire and is committed to making West Ham a great Premier League club," Gold and Sullivan said in the letter published on the club website (www.whufc.com).
"These may be tough times for everyone at West Ham but no one at Upton Park is panicking yet. Sullivan and Gold mean what they say. Both of them believe, and have believed all season, that Allardyce is the best man to keep them up.
"Premier League survival is paramount. The club have done all the figures on staying in the top flight and will spend again in January with that in mind."
The Hammers have been badly hit by a series of injuries with record signing Andy Carroll sidelined all season because of a foot injury.
Gold and Sullivan said West Ham were close to strengthening their squad in the transfer window.
"January is not an easy time to do business but we are pleased to confirm we are close to securing one of our key striking targets," the pair said.
"A defender is high on the list of priorities as well and progress is being made on that front."
Allardyce, who has been at Upton Park since June 2011, said he was forced into fielding a young and largely inexperienced side at Forest, making nine changes from the team beaten 2-1 at Fulham on New Year's Day.
"Many key players are out injured and Sam has been handed a near impossible task of coming through three crucial games in the space of six days with a squad of only 14 fit and available senior players," read the letter.
"This is not an excuse, it is a fact. We are under no illusions as to the seriousness of our current predicament in the Premier League.
"But on the board we have a combined talent of running football clubs and we will draw on all of our nous to get West Ham United out of this situation.
"We are all working round the clock to do everything in our powers to help arrest our slump in form."