Le Pen, who lost the presidential election in May to Emmanuel Macron, said her National Front (FN) party was being deliberately cut off from financing as part of a "banking fatwa" against the far-right.
"After being the victim of massive judicial persecution, we are witnessing a new stage in the persecution of the National Front -- banishment from banking," she told a press conference.
Her claim of "judicial persecution" was a reference to the decision by parliament earlier this month to strip her of her immunity from prosecution for tweeting pictures of atrocities by the Islamic State group.
Le Pen said that Societe Generale had asked the FN to close its accounts, while HSBC's French boss Thomas Vandeville called her Wednesday to announce her personal account was being shut, "without any justification".
Societe Generale said in a statement that its decisions "on whether to open or close a bank account depend purely on banking reasons and in respect of all regulatory requirements, without taking into account any political consideration".
A spokeswoman for HSBC France said the bank does not "publicly discuss our relationships with our clients."
Le Pen repeatedly complained during the election campaign that the FN had been refused loans by both French and foreign banks.
The party borrowed nine million euros ($10.5 million) from a Russian bank in 2014, prompting critics to question whether Moscow had influence over the party.
Le Pen said Societe Generale's recent decision was "depriving a party that won 11 million votes in the last presidential election of all practical ability to function".
"We are cut off at present from our income. This decision puts the National Front in a position of serious difficulty and prevents the party from functioning normally," she said.
"We are witnessing an attempt by the opposition to suffocate us."
She blasted Societe Generale's decision as a politically motivated measure against a party which has "tens of millions of members, stable resources and no problems with our accounts whatsoever".
Le Pen said she had raised her party's financial problems with Macron.
She became a personal customer at international giant HSBC after her bank Hervet was taken over in 2001 by Credit Commercial de France, which is part of HSBC.