LONDON (Reuters) - A judge has given Royal Bank of Scotland RBS.L and thousands of investors until June 1 to agree a deal that would avert a trial over the bank's 12 billion pound cash call in 2008.

RBS has already offered almost 1 billion pounds ($1.3 billion) to settle investor allegations that it misled them. If it reaches a deal, RBS will avoid a trial that would rake over its near collapse and force former chief executive Fred Goodwin to testify.

Judge Robert Hildyard on Wednesday adjourned proceedings for the third time to June 7 after days of intense talks failed to yield an out-of-court settlement. He ordered both sides to notify him of any deal by June 1 or face trial.

“We must have certainty one way or another ... and that must be made clear,” Hildyard told London’s High Court. The trial had been due to begin on Monday.

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

Jonathan Nash, a lawyer for the claimants, said that progress “remains good” and he was hopeful of a settlement.

Nash blamed logistical problems reaching “a small number” of shareholders, some of whom are elderly, for the delay and said some of their addresses did not appear to be correct.

Sources told Reuters late on Tuesday that a number of investors remained determined to take the case to trial, partly to see Goodwin held to account for decisions which RBS’s current management says were responsible for losses it suffered.

A spokesman for the investor group did not respond to requests for comment. 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 around 200 million pounds - remains well below the 200 to 230 pence 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 includes thousands of current and former RBS employees. It alleges former executives hid over-stretched finances and failed to disclose that the regulator had ordered RBS to raise cash in 2008.

Months later, the government stepped in with a record 45.8 billion pound bank bailout. Shareholders lost around 80 percent of their investments - and some their livelihoods as RBS cut jobs as part of a major restructuring.