But as it sent out letters and documents detailing the final settlement for all 34,000 members of the market - who will receive their individual statements over the next few days - Lloyd's insisted it was optimistic that the 2,700 American names could be brought into the deal.
The letters began the last stage of plans to rescue Lloyd's and put it on a sound footing, with a deadline of 28 August now set for the rescue package to be accepted by the market's members.
The settlement offer said among the difficulties in the US were legal claims being made that the establishment of Equitas, the reinsurance vehicle into which the market's old liabilities have been lumped, involves the issue of a security, bringing it into the remit of securities regulators in the US.
Lloyd's disputes this challenge, but says it could have a serious impact on its ability to include all US names in the settlement offer.
Ron Sandler, chief executive, says in an introduction to the document: "Progress has been made in recent weeks but I cannot rule out the possibility that we may decide not to make the offer to US names, either generally or in certain states."
Mr Sandler tells the names that if the offer is not accepted there is a real risk of a failure of Lloyd's, which has lost pounds 8bn in the last five years. If that happened, names and their estates would have full liability to their policyholders, and mechanisms for debt collection would "operate with full force".
British names' representatives welcomed the improved offer and said it was likely to receive a positive response. There is a wave of opposition by names' action groups to attempts to change the deal at an extraordinary meeting of members next month.
One of the harshest critics, Christopher Stockwell, of the Lloyd's Names Associations' Working Party, which called the EGM, said: "What we have got is a big improvement on the offer that was made initially.
"This is a way that gives us a solution now. I anticipate that a substantial majority will wish to accept, although a minority of names will feel it is not good enough."
David Rowland, Lloyd's chairman, in a letter to names accompanying a detailed information package, said he deeply regretted the events that had made the reconstruction plan necessary.
"A very supportive, very encouraging response from the members," was how a Lloyd's spokesman described a meeting of 400 Australian names with deputy chairman John Stace this week.
Before taking their funds at Lloyd's into account, 12,000 names will receive a release from additional payments, 6,800 will have a bill of no more than pounds 25,000 pounds, 4,100 will have to pay pounds 25,000-pounds 50,000,while 3,600 will have to pay pounds 50,000-pounds 75,000.