UPDATE 1-Judge gives RBS week to end investor suit or face trial

* Former RBS CEO Goodwin scheduled to testify in June

* Bank has settled with majority of claimant groups (Recasts, adds details)

LONDON, May 24 (Reuters) - A British judge has given Royal Bank of Scotland a week to avoid a trial by reaching a deal with investors who allege the bank misled them over its 2008 fundraising.

Judge Robert Hildyard adjourned the case on Wednesday until June 7, but warned this would be the final chance to reach an out-of-court settlement and said the two sides must inform him whether a settlement has been reached by June 1.

RBS has already offered almost 1 billion pounds ($1.3 billion) to avoid a trial that would rake over its near collapse and state bailout during the height of the credit crisis and bring former chief executive Fred Goodwin to court.

The trial, which was due to start on Monday, has been adjourned to give the parties time to negotiate an 11th-hour offer from RBS aimed at ending the lawsuit.

RBS, which is still more than 70 percent state owned, denies any wrongdoing and says former executives did not act illegally.

"We must have certainty one way or another ... and that must be made clear," Hildyard told London's High Court.

Jonathan Nash, a lawyer for the claimant group, told the court that "progress remains good" and he is hopeful that shareholders who are so far holding out will agree to a deal.

Sources told Reuters late on Tuesday that a number of shareholders in the 9,000-strong group suing RBS remained determined to reject its offer and take the case to trial.

A spokesman for the investor group did not respond to requests for comment, while RBS declined to comment.

RBS almost doubled its out-of-court offer to investors on Sunday from around 43.1 pence to 82 pence per share, sources told Reuters. The offer - set to cost RBS tens of millions of pounds - is below the 200 to 230 pence per share at which shareholders bought RBS shares in 2008.

Investors representing 87 percent of the original 4 billion pound damages claim have already settled..

But the remaining group, which includes thousands of current and former RBS employees who lost around 80 percent of their investments, alleges former executives hid over-stretched finances and failed to disclose that the regulator had ordered RBS to raise cash when it launched its rights issue in 2008.

Months later, the government had to step in with a 45.8 billion pound bailout, the largest bank rescue in history. ($1 = 0.7702 pounds) (Additional reporting by Lawrence White; Editing by Rachel Armstrong/Mark Potter/Alexander Smith)