The suit, filed in a Manhattan court by Andrea Barron, accuses UBP and executives at its asset management arm of breaching their fiduciary and other professional duties by "failing to perform adequate due diligence" on Mr Madoff or his firm, despite numerous warnings about the broker, including some from its own research team, Reuters reports.
According to the news agency, the Geneva-based bank's total exposure to the $65 billion Madoff scandal was more than 1 billion Swiss francs ($904 million).
In March, the institution offered to buy back its clients' exposure to Mr Madoff and Bernard L Madoff Investment Securities at 50 percent of their original cost.
Commenting on the lawsuit, the bank said: "The complaint is without merit and UBP will defend it vigorously."
Reuters said the suit is seeking class-action status for damages, as well as the return of fees earned by UBP.
Mr Madoff pleaded guilty to charges including securities fraud, money laundering and perjury in March.