The Portuguese coach is facing mounting pressure as the crisis worsens at Old Trafford.
A fractured pre-season, claims of dressing room unrest, poor business in the transfer window and two defeats already this season mean Mourinho is already facing calls to leave United in the coming weeks.
Defeat to Burnley this weekend would heap further pressure on the 55-year-old with executive vice-chairman Ed Woodward growing increasingly concerned.
And Allardyce says at a price of 100/1, he would back himself to replace Mourinho at United this season.
On talkSPORT this morning, the former England boss was asked by Alan Brazil: “I was going to ask you, you’re in the running at 100/1. Can we have a pound on you?”
To which Allardyce quipped: “I’ll have a pound on myself.”
The 63-year-old has been out of work since leaving Everton last season after fans were unhappy with the club’s style of play.
As the odds suggest, Allardyce is unlikely to be drafted in should Mourinho face the axe in the near future.
At present, Zinedine Zidane is the bookmakers’ favourite to replace Mourinho at odds of 7/4, while Antonio Conte and Mauricio Pochettino can be backed at 8/1.
Massimiliano Allegri and Carlo Ancelotti, both currently managing in Italy, are priced up at 10/1 and 16/1 respectively.
After the 3-0 humbling at home to Tottenham on Monday evening, some United fans took to social media to demand Mourinho’s sacking.
“No words…….absolute shamble #MUFC #MourinhoOut #MANTOT @ManUtd @SpursOfficial,” one fan wrote.
Another said: “Get Mourinho out of @ManUtd!!!”
“Won’t even be anywhere near top 4 this year,the players don’t want to even play for the club,shocking to watch, Mourinho Out,” a third fan wrote.
A fourth supporter said: “Enough now Mourinho, pick up your p45 on the way out.”
Mourinho is unlikely to walk away from one of the best-paid jobs in world football, while it will reportedly cost United £50m to pay the remainder of the Portuguese’s contract.