A group of Names has confirmed it is to pursue fresh legal action against Lloyd's.
This follows Lloyd's winning the Court of Appeal case against the Jaffray Names in July.
The court found Lloyd's not guilty of fraud, but found it had made certain misrepresentations to investors.
United Names Organisation (UNO) co-chairman Catherine Mackenzie-Smith said:
"We have issued an application to amend our pleading in the light of the Court of Appeal finding to the effect that during the period 1978-1988 Lloyd's made misrepresentations."
She said they are also applying to extend the time period as some of the UNO Names joined Lloyd's after 1988.
If the Names are successful in the new action they could be eligible for the same amount of compensation as they would have been if they had won the fraud case.