Miguel Herrera has been fired as Mexico coach following allegations that he punched a television reporter.
Former Arsenal midfielder Abou Diaby has rejected an offer to join West Brom in favour of finding a new club in France.
Roberto Di Matteo had been considered the most likely candidate to replace his compatriot Paolo Di Canio but has now drifted out to 9/1 to get the job.
A consistent run of bets on Poyet has seen him come in from 8/1 on Tuesday morning to his current price of 4/6, while Steve McClaren (9/1), Tony Pulis (10/1) and Gianfranco Zola (14/1) are considered alternative options.
The Uruguayan boss is out of work since leaving Brighton over the summer, although he is still embroiled in a legal battle over his sacking.
Sky Bet Football Trader Chris Spicer said: "We installed Di Matteo as the initial favourite and our punters seemed in agreement as he attracted the majority of the bets, but there has been a significant shift in where the money is going.
"While we've seen some interest in McClaren, Pulis and Zola, the majority of the stakes are going on Poyet."