Cameroon international Bassong complained to referee Howard Webb during Saturday's 4-3 win at Swansea City and a man in the crowd was arrested and later charged in relation to alleged racist abuse.
"Norwich 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 statement said.
Norwich said a tweet containing racist abuse was also sent to Bassong's twitter account after the game and details have been passed on to Norfolk police who are also investigating tweets sent to the defender following the 1-1 draw at Everton last month.
"Sebastien Bassong is a fantastic athlete and professional who we are proud to have on board at Norwich City," the club said.
"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."