Both South Wales Police and Norfolk Police are now investigating the claims.
A Swansea supporter was released on bail on Sunday after being arrested during Norwich's 4-3 victory on Saturday for allegedly making racist gestures towards Bassong. It is believed the 23-year-old had made gestures towards the player as Norwich celebrated their fourth goal in the game and he was arrested soon after the incident.
Norwich have confirmed the other three cases involved racist messages being sent to the defender on Twitter after both 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."
The club also thanked both Swansea and the police for their treatment of the situation and also confirmed racist Tweets aimed at Bassong have also now 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."
Source: PA
Source: PA