The incidents relate to City's 4-3 win against Swansea at the Liberty Stadium on Saturday and the 1-1 draw at Everton last month.
A Swansea fan was released on bail on Sunday after being arrested at the game for allegedly making racist gestures towards the Cameroon international.
The 23-year-old supporter is understood to have made gestures towards Bassong as the Canaries celebrated their fourth goal and he was arrested soon afterwards.
"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," read a statement on Norwich's official website.
"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.
"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.
"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."
Kick It Out chairman Lord Ouseley praised the way in which Saturday's incident had been dealt with.
"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."
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