ZURICH (Reuters) - Swiss lawmakers on Wednesday voted to relax strict secrecy laws to make it easier to liquidate unclaimed accounts held in Swiss banks, a major step forward in efforts to deal with wealth hidden by Jews from the Nazis during the Second World War.

Though Switzerland's UBS and Credit Suisse reached a $1.25 billion deal with Holocaust survivors and their descendants in 1998, the Swiss government has spent years attempting to amend banking secrecy laws to try to deal with unclaimed accounts.

On Wednesday, the lower house of parliament made a breakthrough, agreeing to suspend secrecy rules in order to allow unclaimed funds to be moved to another bank and liquidated without an account holder's approval.

The secrecy rules would only be waived if the bank could prove it had repeatedly failed to reach the account's owner or their beneficiaries.

Switzerland's upper house has yet to vote on the changes.

The 1998 settlement followed three years of wrangling between Holocaust victims and their descendants and Swiss banks over the fate of funds deposited in Swiss banks by Jews during the Second World War.

Swiss banking secrecy has come under renewed pressure in recent years, forcing Switzerland to relax its banking laws in certain areas and pledge to pursue tax evasion as well as outright fraud.

Switzerland is currently grappling with a U.S. crackdown on wealthy Americans hiding funds from the Internal Revenue Service after U.S. prosecutors indicted Swiss private bank Wegelin over allegations it had helped hide money.

Switzerland is also seeking to head off European Union pressure to make its banking system more transparent with a series of withholding tax deals and lump-sum payments.

Swiss officials recently pledged a clean-money strategy, and are now seeking to ensure that private bankers do not deal with funds that have not been taxed.

(Reporting By Katharina Bart; Editing by Andrew Osborn)