West Ham have plummeted to second-bottom in the top flight after winning just one of their last 13 league games and they were thrashed 5-0 at Championship outfit Nottingham Forest in the third round of the FA Cup on Sunday.
Allardyce was forced to field a youthful side at Forest due to the club's well-documented injury crisis, and with the first leg of their League Cup semi-final at Manchester City on Wednesday in the back of his mind.
Harry Redknapp is odds-on favourite to return to Upton Park, but West Ham co-owners David Gold and David Sullivan issued their support of their manager to fans in an open letter on Monday and the former Blackburn Rovers boss is grateful for their backing.
"I'm obviously pleased with the united front we're showing in these circumstances," Allardyce said on Tuesday.
"We all understand the difficult position we're in. I have a lot of sympathy for the fans at the moment."
Allardyce has urged his side to be resolute in the first leg at the Etihad Stadium on Wednesday against a rampant City side.
And the West Ham boss says his squad will confidence from a 2-1 quarter-final triumph at Tottenham, when Matt Jarvis and Modibo Maiga scored in the last 10 minutes after they weathered a storm at White Hart Lane.
"I have to make available the freshest team I could for Wednesday against the most free-scoring team in the Premier League," he said.
"We have to be defensively resilient as we were at Tottenham and then we'll have a chance to do well again on Wednesday.
"We don't have to win but we have to come back here with something realistic. We have to try to nullify a great attacking force.
"We are in an exciting cup tie with a massive joy at the end of it if we get to Wembley.
"We need a bit of fortune and to be organised and committed and who knows what can happen?"
Allardyce is hopeful that Mark Noble will return from a calf injury on Wednesday, and revealed striker Andy Carroll is not far away.
That will not stop him looking bolster his squad in the January transfer window though.
"Of the injured players, Mark Noble has the biggest chance to be fit to play tomorrow and that would be a big boost for us," he added.
"Andy Carroll is training today and hopefully we'll see him on the bench in the next week or so.
"(As for the transfer window), we're reasonably close to bringing in another player, but we never know until the deal is signed. We hope one or two may get clinched."
West Ham signed Wolves defender Roger Johnson on loan on Monday until the end of the season.
