The company's case is expected to argue that the banks have "neglected" victims of the alleged fraud, City AM reports.
UBS, which is Switzerland's biggest bank and a global player in asset management and investments, said it did not recommend investing in Mr Madoff's products, the site noted.
However, it did establish a fund in Luxembourg, Luxalpha, that did invest with the Wall Street broker, the bank said.
In December, London-based HSBC confirmed it had a number of custody clients who had money invested with Mr Madoff.
The bank, which has offices in 85 countries, also said it had provided financing to a "small number of institutional clients" who were exposed to the scandal.
In total, HSBC estimated its potential losses from the affair at $1 billion.