Budget Minister Jérôme Cahuzac said he will sue investigative news website Mediapart over the allegation that he held the account with UBS until 2010.
Prime Minister Jean-Marc Ayrault stood behind Cahuzac, saying there was no question of him leaving the government.
"I have every confidence in Jerome Cahuzac and I can imagine what he is feeling right now," Ayrault told RTL radio.
"He has started to fight back, it's difficult, but he will fight for his honour."
President François Hollande, other ministers and the Socialist Party have all given their support to Cahuzac, who described the allegations against him as "crazy".
Even the right-wing opposition UMP has backed him, with party chief Jean-Francois Copé expressing his "personal esteem" for Cahuzac.
"There is no tangible evidence," former UMP minister Bruno Le Maire said.
"We should not, on the right or the left, take part in a witch-hunt against the budget minister,"
The allegations are bound to be troubling, however, for the government of Hollande, who made ministers sign an ethics code and vowed to rule with integrity when he took office in May.
A wealthy former plastic surgeon, Cahuzac has been leading efforts to fight tax evasion as the Socialists seek to eliminate France's budget deficit and revive the country's struggling economy.
Last month he launched an anti-evasion campaign aimed at bringing in a billion extra euros in tax revenues next year.
Médiapart says its report was based on several witness accounts and documents showing that Cahuzac owned the Swiss bank account, undeclared to French tax authorities, for several years before transferring the funds to a Singapore account in 2010.
The news site released an audio recording reportedly of Cahuzac admitting to having the account. In the recording, a man is heard saying: "It bothers me to have an account there, UBS is not necessarily the most hidden of banks."
The recording was left on an answering machine in 2000, when Cahuzac reportedly pressed redial on his phone by accident and then had the conversation with someone in his office.
Médiapart said it confirmed the information with a former tax official, who had raised the existence of the account in a memo to his superiors in 2008, and with a source at UBS.
In a statement after the report was released, Cahuzac said the evidence presented by Mediapart was neither "impressive" nor "convincing".
"I do not have, and have never had, a bank account in Switzerland or elsewhere abroad," he said.
"No credible witness can assert or try to corroborate something that does not exist and has not existed."
Médiapart is known for its investigative reporting and in 2010 broke the story of a campaign financing scandal involving ex-president Nicolas Sarkozy and L'Oréal heiress Liliane Bettencourt.
Sarkozy was questioned for 12 hours last month by judges investigating the allegations but was not charged.
Little-known before being named a minister, Cahuzac, 60, began his career as a cardiologist before switching to the more lucrative world of plastic surgery, in particular hair transplants.
Elected to the National Assembly in 1997, he headed its commission on public finances and was named budget minister when Hollande formed his first government.
French media reported in October that Cahuzac's 300-square-metre (3,230-square-foot) apartment in Paris's exclusive 16th district was robbed, with thieves taking 100,000 euros ($130,000) worth of luxury watches.