Garry Monk's future as Swansea boss is increasingly uncertain, but BBC Wales Sport understands David Moyes does not want the role if it becomes available.
One bookmaker has stopped taking bets on Monk being the next Premier League manager to be sacked.
The Swans have won once in 11 league games after the 3-0 loss to Leicester.
Moyes, 52, was sacked by Real Sociedad in November and is believed to be waiting for the right opportunity to return to management.
Having enjoyed a successful 11-year tenure at Everton, the Scot lasted less than a season at Manchester United before moving to Spain in November 2014.
Swansea have yet to decide on the future of Monk, who succeeded Michael Laudrup in February 2014, initially on a temporary basis.
However, William Hill have suspended betting after what they called a "rush of ... wagers on Gerry Monk over the course of 10 minutes on Monday morning."
The 36-year-old led the Swans to eighth place last season, their highest Premier League finish in his first full campaign.
This season has been startlingly different, however, and Swansea's defeat by Leicester was their seventh in 11 league fixtures with a trip to Manchester City due next Saturday.
Monk said in November he has the "full backing" of chairman Huw Jenkins, while captain Ashley Williams has voiced his support for the former Swans skipper.
But recent results have caused concern, prompting the Swansea board to consider Monk's position.