Snodgrass turned down a new long-term deal with the Tigers last month and although the club recently triggered his one-year contract extension, West Ham still tabled a reported offer of £3million for the Scotland star.
A club source has been quoted as saying the bid had been rejected but West Ham are expected to further test Hull's resolve, while the 29-year-old's refusal to commit himself to the KCOM Stadium has alerted several other Premier League clubs.
Hull sacked boss Mike Phelan on Tuesday night and have wasted little time in appointing Portuguese Marco Silva as his replacement.
Silva was introduced to the media at the club's Cottingham training ground on Friday and West Ham appear to have been ready to take advantage of the upheaval.
Snodgrass has been a stand-out player for Hull this season as they struggle to adapt to life back in the top flight.
The former Leeds and Norwich midfielder missed most of the 2014/15 season and half of the following campaign due to a serious knee injury.
But he has rediscovered his best form with the Tigers this season, scoring in the recent home draw against Everton with a brilliant free-kick - he had hit the crossbar with a similar effort 10 minutes earlier - and also found the target in Monday's defeat at West Brom.