Royal Bank of Scotland plc  (RBS) shares surged Tuesday after a report suggested the British bank is on the verge of agreeing a multi-billion settlement with a U.S. regulator over the sale of mortgage backed securities during the financial crisis.

Britain's Sky News reported that RBS and the Federal Housing Finance Agency (FHFA) are nearing a settlement that could cost the bank as much as £3.5 billion ($4.5 billion), enabling the bailed-out lender to put some of its mortgage backed securities legacy behind it.

However, a settlement with the FHFA will not mean that RBS is completely out of the woods yet as it must still face talks with the U.S. Department of Justice, which are expected to cost it substantially more than any deal with the FHFA.

RBS shares rose more than 3% during early trading in London, to change hands at 257.9 pence, outpacing the 0.70% gain of the Stoxx Europe 600 Banks index.

Briefly the world's largest bank by assets, RBS is currently the world's largest surviving underwriter of mortgage backed securities, having taken second place only to Lehman Brothers and Bear Stearns ahead of the financial crisis.

Accordingly, it has widely been expected to take the biggest hit from U.S. regulators in settlements over allegations of misselling, which have hung over much of the banking industry during the years since the crisis.

Deutsche Bank AG  (DB - Get Report) settled with the Department of Justice for more than $7 billion in December while other lenders, including Credit Suisse Group  (CS - Get Report) , have also done deals in order to put the past behind them.

Recently, RBS reached a landmark agreement with a group of 27,000 shareholders who sued the bank, claiming that it misrepresented its financial position when it set out to tap investors for £12 billion back in April 2008, just months before Lehman Brothers collapsed and the financial crisis began to snowball.

The settlement cost RBS around £1 billion. In October 2008, a month after the collapse of Lehman Brothers, RBS was bailed out by the U.K. government in a £45 billion ($58.5 billion) rescue that saw the taxpayer take what is now a 70% stake in the bank.