UBS was also named as an "assisted witness" in an investigation into a money laundering operation that was part of a tax fraud, the source said on Friday.
The alleged offences date back to the 2000s, the source said.
Investigators suspect UBS allowed its Swiss staff to illegally approach French clients and set up a shadow accounting system to hide movements of capital between France and Switzerland.
A former head of UBS France, Patrick de Fayet, a former head of the bank's office in the northern French city of Lille and a UBS executive in the eastern city of Strasbourg are already under investigation.
The French probe was launched after allegations from a former UBS employee turned whistleblower.
An anonymous note seen by AFP that was sent to the ACP, the Bank of France's regulatory arm, alerted the body to the parallel accounts that ran between 2002 and 2007.
These listed accounts opened in Switzerland by businesses but not declared in France.
The investigating magistrates handling the UBS affair have sent a list containing 353 names of people suspected of having held a Swiss account and have requested details from the Swiss authorities.
The issue jumped back to the top of the government's agenda in the wake of a scandal surrounding the former budget minister Jérôme Cahuzac who in April was himself placed under investigation for tax fraud.
After months denying the allegations, Cahuzac admitted to opening an undeclared Swiss bank account in 1992, and, after Switzerland pledged to cooperate with foreign tax authorities in 2009, transferring the some €600,000 ($770,000) to Singapore.
His role as budget minister included tackling tax evasion.