The Swiss National Bank would not jump to the rescue of any banks that found themselves insolvent following fines to settle U.S. tax evasion probes, the central bank's chairman said.

"It is not the National Bank's job to save banks if they are insolvent. It has a duty to contribute to the system's stability," Thomas Jordan told the Schweiz am Sonntag newspaper in an interview.

The Swiss government is trying to push through a law that will allow Swiss banks to hand over internal information to U.S. authorities in the hope of avoiding threatened U.S. criminal charges over allegations that they helped wealthy Americans hide money from the tax authorities.

The banks are expected to face fines likely to total billions of dollars.

In 2008 the SNB helped bail out systemically relevant UBS . Asked whether the situation was different with smaller banks, Jordan said: "The National Bank's mandate is clear: we have to contribute to the system's stability."

Banks under formal U.S. investigation include Credit Suisse , Julius Baer , British bank HSBC's Swiss arm, privately held Pictet in Geneva and smaller players such as LLB's Swiss arm and local government-backed Zuercher Kantonalbank and Basler Kantonalbank .

Jordan also warned about the risk of a housing bubble, despite government measures announced in February to dampen a housing market boom fuelled by ultra low interest rates.

"We're observing that some banks are starting to change their behavior. However as a whole we are still seeing an inadequate flattening of the dynamic," Jordan said.

He said a big risk was the number of mortgages given to people that might be pushed to their limits in the long-term.

"It would only need a small change in prices or interest rates for these households to get into problems," he said.

Jordan reiterated that the franc remained highly valued, despite having weakened below 1.25 last month.

Asked whether the SNB would use the recent weakening to move its 1.20 cap on the franc, which it set in September 2011 to prevent deflation and recession, Jordan said the central bank was open to all possibilities.

"We are not ruling out in principle any measures that help us to guarantee adequate monetary conditions. However, I must stress that we are not conducting a fine-tuning policy," he said.

(Reporting by Caroline Copley)