Anti-racism organisation Kick It Out has contacted Leicester after allegations concerning striker Jamie Vardy.
The Sun on Sunday published a video allegedly showing Vardy abusing a fellow gambler in a casino and calling him "Jap" on three occasions.
The incident is reported to have taken place in the early hours of July 26.
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Leicester were unavailable for comment when contacted by Press Association Sport on Sunday, but Kick It Out has already opened dialogue with the club.
Kick It Out director Roisin Wood told The Sun on Sunday: "It is disappointing that the opening weekend of the season, which should be a time of celebration, has been overshadowed by these allegations.
"Footballers, no matter the level they play at, are role models on and off the field, and their behaviour can impact millions of fans across the country.
"If these allegations are proven, we would consider it to be a serious matter and would expect the club to conduct a swift and thorough investigation. We have contacted the club and await their response."
Vardy opened the scoring for Leicester in their 4-2 opening day victory over Sunderland.
Leicester are owned by the Thai Srivaddhanaprabha family and it is only a matter weeks since the Foxes dismissed Tom Hopper, Adam Smith and James Pearson, son of former manager Nigel, for their conduct on a close season tour of the country.
The trio had featured in a sexually-explicit video during which racist language was used and had their contracts terminated following an internal investigation.
A statement released by the club at that time read: "Leicester City Football Club is acutely aware of its position, and that of its players, as a representative of the city of Leicester, the Premier League, the Football Association and the club's supporters. It is committed to promoting a positive message of community and family values and equality, and to upholding the standards expected of a club with its history, tradition and aspirations."