Pulis won the LMA's Premier League Manager of the Year award last season for an impressive six months in charge in which he replaced Ian Holloway mid-season and pulled the club out of relegation danger.
Under Welshman Pulis' watch, Palace went from strength to strength, finishing 11th after a five-match winning streak - with only one loss in their last eight games.
His position was seen as one of the safest in the division and at the start of Thursday he was 40/1 outsider to be the first manager to leave.
However, a host of bets has seen him made the new favourite with all bookmakers, with some offering just even money on Palace being the first club to make a change.
According to the Daily Telegraph, Pulis could leave in the next 24 hours as he is unhappy with the transfer funds that have been allocated to him this season.
Pulis wanted to sign both Steven Caulker and Gylfi Sigurdsson this summer but the players instead went to QPR and Swansea respectively.
Co-chairman Steve Parish comments regarding Pulis's future will have done little to dampen the speculation.
According to the Evening Standard, when asked if Pulis was still manager, Parish said: I don’t know, I’m doing something completely different. I can’t comment. I don’t know anything about it.”
The paper also reports that when Parish was pushed on whether he could guarantee that Pulis would be in charge when Crystal Palace take on Arsenal on Saturday, Parish was non-committal: "I really don’t know anything about it.”
The BBC and Sky Sports were among various outlets reporting that Pulis and Parish were meeting on Thursday evening.
Earlier in the summer Pulis said it was important that the club built on their success last season.
"This is a community club based in the most densely populated area in London and has a great opportunity of pushing forward," he said on Palace's website.
"We've got to stay in the Premier League for the next two or three years to realise what we can eventually achieve."
Willham Hill said they had no option but to slash their odds after a "monster" gamble on Pulis leaving.
"This is a crazy gamble,” said William Hill spokesman Joe Crilly.
"His odds went from 40/1 to 7/2 this morning and despite being an incredibly short price punters are still backing him. We started to see activity on the South Coast early this morning and now the gamble has snowballed."
Pulis replaces West Ham boss Sam Allardyce as the favourite to leave first, with Aston Villa's Paul Lambert third in the betting.
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