The Orange Order has accepted an invitation to meet the British Prime Minister, Mr Blair, in Downing Street on Thursday, but has warned that it is unlikely to change its mind and accept the Belfast Agreement.
The Order had requested the meeting to discuss the agreement, but sources have expressed annoyance about the time it took to arrange a date.
The chief executive of the Grand Lodge, Mr George Patton, said: "None of us are going to London with closed minds, but I think it will take an awful lot of convincing. This is a green document and there is nothing in it for unionists.
"I have my doubts that this meeting will make any impression. The vast majority of Orangemen are opposed to this agreement. I have already had enough clarification after listening very carefully to what has been said in Parliament and other places."
Other Orange Order sources said they agreed with Mr Patton's position, but Grand Lodge secretary, Mr John McCrea, said the Order would listen to Mr Blair. "If he tells us something that will change our minds, then we will take all aspects of his comments into account."
The Orange Order opposes the agreement because of the proposed early release of paramilitary prisoners; the possibility of Sinn Fein holding executive positions without prior IRA decommissioning; and fears that the RUC will be disbanded or radically changed; and because the Republic will have a major role in the North's internal affairs.
However, UUP deputy leader, Mr John Taylor, urged the Order not to be negative. "Many, many Orangemen are voting Yes and it is terribly important that the leadership does not find itself out of step with the greater number of pro-British people in Northern Ireland.
"It's important people appreciate that all of us have problems with this agreement. I have 18, but advances have been made on the constitutional issue and that is vital in guaranteeing our future as part of the UK."
Another UUP MP, Mr Ken Maginnis, also said the Orange Order leadership was in danger of becoming out-of-touch with grassroots unionist opinion. "There has not been any objective debates within the Order. It has been hijacked by a comparatively small element and that is sad."
Half of the UUP's 10 MPs are in open revolt against the document and a sixth, Mr Jeffrey Donaldson, has said he personally will vote No.
Mr Blair is due to visit the North later this week, but dissident UUP MP, Mr Willie Ross, said his trip would make no impact. "There is a huge groundswell of people opposed to this agreement and the vote will be much closer than people think. It might even go No," he said.
Another dissident UUP MP, Mr Willie Thompson, said his party leadership had made "a gross error of judgment" and claimed opposition was growing.
"No party can expect blind loyalty and maybe now the men in grey suits will see sense. The No campaign is gathering momentum. We may not win it because of the nationalist vote, but what we will see is a majority of Protestants will reject it. That will produce a majority of No unionists in an Assembly, enough to kill it off."
Some of the rebel MPs have joined a `united unionist campaign' with the DUP and the UK Unionist party and a special headquarters is to be set up.