The organisations - Sherpa, which defends victims of economic crimes, Ibuka France, a Rwanda victims' association, and the France-based Collective of Civil Parties for Rwanda - accuse the French bank of financing the purchase of "80 tonnes of arms used to carry out genocide" by the Hutu regime.
The four-month campaign led to the massacre of about 800,000 people, mostly from Rwanda's Tutsi minority.
BNP authorised the fund transfer to the regime in June 1994, the groups said in a statement, accusing the bank of "complicity in genocide, war crimes and crimes against humanity".
According to the groups, "the bank had to have known the genocidal intentions of the country's authorities."
A spokesman for the bank told the AFP news agency it could not comment on the claims, saying it did not yet have "sufficient details".
To read our reports Rwanda remembers genocide 20 years later click here