McCaren was told he had two games to save his job, but Newcastle's win over Liverpool on Sunday saw an improved performance on recent weeks.
Would Rodgers take the Toon job? Insiders suggest he would ... if he was given full control over team matters, unlike McClaren.
Dennis Bergkamp was being backed in at Sky Bet on Thursday morning to become Swansea's new manager.
The former Arsenal striker was originally a 28/1 shot when the market opened in the wake of Garry Monk's sacking on Wednesday afternoon.
He had been cut into 14/1 but Wednesday night but on Thursday morning the price shortened further to 8/1.
Sky Bet's Zach Amin said: "Dennis Bergkamp has been the surprise choice so far from the punters and he is now 8/1. Bergkamp is currently learning his trade as the Ajax assistant manager and would be seen as a risky choice if chosen for the job."
The 8/1 price still made him only fourth favourite for the Liberty Stadium post, though, with former Everton and Manchester United boss David Moyes the favourite at 2/1 despite reports that his agent had said he was not interested in the job.
Gus Poyet was being better supported and was cut into 5/2 from 7/2 on Thursday morning after some chunky bets, while former Swans chief Brendan Rodgers was on offer at 6/1 about a return to the role.
Rodgers, axed by Liverpool earlier this season, had gone odds-on in the hours following Monk's departure but reports suggested he was not under consideration by the Welsh club.
Monk was sacked following a run of one win in 10 Premier League games, culminating in Saturday's 3-0 home defeat to Leicester.
He leaves the club in 15th place in the table, just one point outside the relegation zone.