The daily Le Parisien reported that both UBS and Credit Suisse were asking for a document signed by a lawyer or an accountant "allowed to practise in France" certifying customers' accounts are in line with tax authority requirements.
The paper quotes a UBS document as saying that "if the assets held at UBS have not been declared to relevant tax authorities", the customer will be asked to "consider measures necessary for regularisation."
Switzerland is famed for its bank secrecy laws, which critics say have enabled many clients to shield their wealth from the scrutiny of tax inspectors back home.
But these once sacrosanct laws are slowly being eroded as France and other countries such as the United States step up their fight against tax evasion in a troubled economic climate.
On Monday, Swiss MPs voted in favour of a law requiring Swiss banks to report the holdings of their US clients to US tax authorities in line with the US Foreign Account Tax Compliance Act, which aims to ensure that all US citizens are taxed on their income and assets worldwide.
France has also been cracking down on tax evasion following a scandal involving former budget minister Jerome Cahuzac, who was forced to resign in disgrace earlier this year over an undeclared account.
Le Parisien quoted a Swiss banker as saying that most banks in Switzerland would soon "have to do the same thing" as UBS and Credit Suisse.
According to Daniel Lebegue, head of the France division of anti-corruption watchdog Transparency International, Swiss bankers are slowly realising that "banking secrecy will soon disappear."