Leading Names - the individual investors who have traditionally supported the market - admit they are pessimistic about the chances of success. They have refused to pay money owed since October 1996 to help fund the establishment of Equitas on the grounds they did not agree with it.
Lloyd's has been forced to take out a pounds 300m syndicated loan to finance Equitas, the reinsurance vehicle into which its liabilities prior to 1992 have been placed.
Lloyd's has already won a judgment on a "pay now, sue later," basis against the Names, and in March Lord Justice Tuckey refused them leave to appeal.
But next week the Names will urge the Court of Appeal to let them be heard. If they win, the case will proceed immediately, but if they lose, Lloyd's will start seizing assets both here and abroad.
A Lloyd's spokesman was confident its arguments were strong, saying: "All the documentation is in place so that we can proceed with debt collection as soon as the court allows."
When the Lloyd's market was restructured two years ago there was pounds 600m outstanding from Names who refused to pay up. So far it has collected pounds 75m but says it never expected to bring in more than pounds 300m of the pounds 600m.