Back-to-back thumpings in cup competitions have left him in the firing line of disgruntled fans.
Manchester City carried on where Nottingham Forest left off as they thrashed the Hammers 6-0 in the first leg of their Capital One Cup semi-final at the Etihad Stadium on Wednesday.
The result, coming just four days after a 5-0 humbling by Forest in the FA Cup, turned up the heat on Allardyce, whose side sit 19th in the Premier League table and have not won in the league since November.
But the manager, who enjoys strong backing from West Ham co-owners David Sullivan and David Gold, has vowed to lift the gloom at Upton Park.
"There are times when it is good and there are times when it is not so good," Allardyce said.
"You have to accept that when it is not so good you have to draw on your strengths and experience to guide the players through the difficult time you are facing.
"Picking them up and getting them ready for the next game is my job.
"At the end of the day, we cannot keep walking around with our heads down."
Another manager in the spotlight is Manchester United's David Moyes, who will bid to avert a fourth straight defeat in all competitions when Swansea visit Old Trafford in Saturday's late kick-off.
United are seventh in the Premier League, 11 points behind leaders Arsenal and in danger of missing out on Champions League qualification.
United were convincing winners in the corresponding fixture with Swansea on the opening weekend of the season, and Moyes would love a similar outcome just days on from losing 2-1 at Sunderland in the first leg of their Capital One Cup semi-final tie.
However, Swansea were winners at Old Trafford only days ago, with Wilfried Bony's late header giving them a 2-1 win over United in the FA Cup last weekend.
Certainly Moyes and his players are going all out to address their problems, with goalkeeper David de Gea revealing that the squad has put in extra practice after training to try to turn their season around.
He told manutd.com: "We frequently stay after training to practice crossing and finishing. We do it especially when there are no matches in the following days and I think the extra training helps me and the other players to improve."
Elsewhere on Saturday, third-placed Chelsea visit Hull at the KC Stadium, Tottenham host third-bottom Crystal Palace and Norwich meet Everton at Goodison Park.
There is a key clash at the foot of the standings as Fulham host rock-bottom Sunderland, while Southampton entertain West Brom at St Mary's.
Second-placed Manchester City are in action on Sunday and will bid to pick up where they left off in their six-goal demolition of West Ham as they visit Newcastle, while on the same day Stoke welcome fourth-placed Liverpool.
Leaders Arsenal are not in action until Monday night at Aston Villa.