The soldiers were on guard in front of the centre as part of reinforced security measures, implemented in the aftermath of the Paris terror attacks last month.
According to reports the man approached the soldiers at 2.10pm and pulled out a big knife, before attacking one of them aiming at his face and his neck.
The other two soldiers were able to overpower the attacker, but one of them was injured in the process.
According to reports on BFM TV, the arrested man had ID papers which carried the name Moussa Coulibaly - the same name of the man who attacked the kosher store in Paris, killing four people last month.
This however is unconfirmed and there is no suggestion as yet that there is a link between the pair.
According to the mayor of Nice, Christian Estrosi, a possible accomplice was arrested nearby.
According to local newspaper Nice Matin the suspect had just been fined for travelling on the tram without a ticket.
"He paid his fine without flinching and then rushed at the soldiers," the newspaper reported.
The centre in Nice where the attack took place was home to the headquarters of the Nice Jewish Consistory - the local Jewish council - and Radio Shalom.
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Following the Paris terror attacks, which included a deadly attack on a Jewish kosher store, the French government deployed thousands of troops to be stationed outside various Jewish venues, including synagogues and community centres.
President François Hollande also vowed to tackle the "unacceptable" rise of anti-Semitic acts in France.
Shortly after the stabbing the mayor of the Nice, Christian Estrosi, tweeted “We are all under attack” and called on the government to ensure the soldiers remained in the city to protect the public.
The anti-terrorist unit of the French police has been put in charge of the investigation.