The money was wanted to help pay for pensions to the hardest hit names who have been wiped out by their losses, which in many cases have forced them to sell their houses and all their assets.
But the extra sum to be contributed by the agents is only half the pounds 50m Mr Rowland had asked for.
The agents resisted his demands by claiming they were already contributing nearer pounds 300m than pounds 200m, after a pounds 64m adjustment for the way profit commission was calculated and a further separate donation they were making of pounds 20m.
The extra cash from the agents emerged 24 hours after Lloyd's signed an outline deal with US securities regulators to top up the value of the rescue package for US names by pounds 40m.
Earlier, Lloyd's said it expected to make almost pounds 3bn profit in the three years to the end of 1995, almost as much as the value of the pounds 3.1bn rescue package which members are to debate on Monday at the annual meeting in the Royal Festival Hall.
The Lloyd's predictions include confirmation of a pounds 1.1bn profit for 1993, the latest completed year, the first time the market had been in the black after five years of losses totalling pounds 8bn. The 1993 profit is after deducting members' personal expenses and a proposed special contribution on which they are to vote on Monday.
For 1994, Lloyd's expects pounds 1bn profit and for 1995 pounds 882m, a lower figure as the decline in insurance rates bites.
Mr Rowland said he expected only a small decline in the number of individual names next year once the rescue goes through and he made clear they would be much more like the members of the 1960s and 1970s, who were richer and more able to withstand financial shocks than many of those who arrived in the 1980s. Lloyd's has just under 13,000 individual names now.
Mr Rowland also made it clear he would be prepared to resign if members voted against contributing pounds 440m to the rescue at the annual meeting on Monday, though he retreated from an earlier off the cuff threat to "head for the hills" immediately if he lost.
Mr Rowland said: "If it does fail, we do not instantly drop the sense of duty we have to the market, but certainly I and the Council would discuss the options with the regulators."
"However, instant resignations and chaos in the market would not necessarily lead to the best outcome. I and my colleagues could possibly continue to give some aid, in whatever form that may be," he added, though he made it clear he would be prepared to resign if asked.
"If this fails there isn't some other magic deal waiting around the corner. There is not an alternative scenario," he said.
Responding to threats of legal actions by members against Lloyd's for fraud, Mr Rowland said: "I think on Monday it will be demonstrated that people crying fraud will be a very small minority."
Meanwhile, the first Lloyd's auction of capacity this year was heavily oversubscribed, and it is expected that almost all capacity will change hands by the new auction system in 1996.Reuse content