Norwich say they have reported four separate incidents of actual or alleged racist abuse of defender Sebastien Bassong to police.
Both the South Wales and Norfolk police forces are now investigating the claims.
A Swansea supporter was released on bail on Sunday after being arrested for allegedly making racist gestures towards Bassong during Norwich's 4-3 victory on Saturday.
It is reported that the 23-year-old made gestures towards the player as Norwich celebrated their fourth goal, and was arrested soon after the incident.
Norwich have confirmed the other three cases involve racist messages being sent to the defender on Twitter after the Swansea match and the 1-1 draw at Everton on November 24.
The club's statement read: "Norwich City regrets to confirm four separate cases involving actual or alleged racist abuse aimed at our defender Sebastien Bassong have been reported to police in the last two weeks.''
Norwich also thanked both Swansea and the police for their handling of the situation and also confirmed that racist Tweets had been reported.
The statement continued: "Following a complaint made by Sebastien to referee Howard Webb during last Saturday's 4-3 Barclays Premier League win at Swansea, a man in the home section of the crowd was arrested and later charged in relation to alleged racist abuse.
"We would like to thank both Swansea City FC and South Wales Police for their swift and effective handling of this unfortunate incident.
"In the evening following the match, a tweet containing racist abuse was sent to Sebastien's twitter account. We have this morning passed details of this incident on to Norfolk Police for them to investigate.
"Meanwhile Norfolk Police are in the process of investigating two further separate racist tweets about Sebastien sent following our 1-1 draw at Everton on Saturday, November 24.''
Norwich also praised Bassong and said: "It is deeply disappointing that his recent excellent performances for the club should be tarnished by these incidents.''
The club's statement said: "Norwich City Football Club wish to reiterate that we have a zero tolerance stance on racism and will pursue these and any other examples of discrimination aimed at any of our colleagues to the fullest extent of the law.''
Information from the Press Association was used in this report.