"Monk was 1/4 to be the next manager to go, yet that did not dissuade punters desperate to back the Swansea manager," said a spokesman for William Hill.
A 3-0 home defeat by surprise league leaders Leicester City on Saturday was the Welsh club's fourth loss in five games and left them in 15th position.
They are now one point above the relegation places.
Monk, 36, is a former Swansea captain who played for the club for 10 years.
He was appointed manager in succession to Michael Laudrup in February 2014.
(Reporting by Steve Tongue; editing by Clare Fallon)
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