Both South Wales Police and Norfolk Police are now investigating the claims.
A Swansea supporter was released on bail 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."
Norwich also praised the player and made their position towards any racist behaviour clear.
"Sebastien Bassong is a fantastic athlete and professional who we are proud to have on board at Norwich City. It is deeply disappointing that his recent excellent performances for the club should be tarnished by these incidents," the statement continued.
"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."
The 26-year-old Cameroon international moved to Carrow Road from Tottenham in the summer and has been an integral part of the club's recent run of good form.
He scored in both games relating to the alleged racist abuse, grabbing a late equaliser at Goodison and heading Norwich 2-0 in front at Swansea.
Norwich boss Chris Hughton believes the speed at which Swansea officials dealt with the fan making gestures towards Bassong should act as a warning to any other people who may consider such actions in the future.
"Swansea City football club have dealt with the incident very, very well and very swiftly and I think that is a way forward," he said.
"On any issues of this kind, and we have seen some issues over the last few weeks, I think if they are handled as quickly as the incident was on Saturday you hope it is a bigger deterrent for anybody who wants to put themselves in that position."
Kick It Out chairman Lord Ouseley also praised the way all parties involved in Saturday's incident dealt with the matter.
"It is a classic, textbook way in which a player who is being abused should complain and how the referee, the officials the security, the police, the club should respond," he told Sky Sports News.
"We want to encourage players to complain, to use the processes, to make sure they work because those processes are there to provide protection and stop people abusing others."
It is understood that Norwich used screen-grabs of two racist Tweets sent to Bassong following the draw at Everton as proof they were sent from accounts that had since been deleted.
Another message sent to the player after the victory at Swansea was also reported to Norfolk Police.