The English sports outlet claims that the bid was in fact made last week and Fulham's board were quick to tell the Premier League team that they were wasting their time and not even an offer of £40m would be considered.
Slavisa Jovanovic's club are in a great position to demand a high asking price for the Scotland international after Cairney signed a contract extension until 2021 with the 'Cottagers' following interest from Newcastle United.
Despite the 27-year-old missing Fulham's 3-1 victory over Barnsley on Saturday with a knee injury, he has been an integral part of the London club's push for promotion this season, making 17 appearances so far.
It will be another reason why Jovanovic would be reluctant to sell Cairney to West Ham with Fulham looking to secure their place back to the Premier League after a four-year absence.