Two British police forces, South Wales and Norfolk, are investigating the complaints and a Swansea fan has been released on bail after being arrested during Norwich's 4-3 victory on Saturday for allegedly making offensive gestures towards the defender.
Norwich have confirmed there are three other cases, all involving racist messages being sent on Twitter after the Swansea match and the 1-1 draw at Everton on November 24.
The club, who have included Bassong in the squad for tonight's League Cup quarter-final against Aston Villa, issued a statement yesterday praising the restraint shown by the former Tottenham player.
"Sebastien is a fantastic athlete and professional who we are proud to have on board at Norwich," it said. "It is deeply disappointing that his recent excellent performances should be tarnished by these incidents.
"The 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."
Lord Ouseley, chairman of 'Kick It Out', the anti-racism group, praised the way Bassong and Swansea had dealt with the matter.
"We want to encourage players to complain, to use the processes and to make sure they work," he said.