The party leader, David Trimble, could, according to some sources, delay revealing his hand until only hours before the formation of a new Northern Ireland government is due to take place on 15 July.
The party says only nominated spokesmen will give media interviews on the key issue of whether or not it will go for the proposed deal. The proposals which emerged from last week's Stormont talks envisage the setting up of the new administration, to be followed within weeks by actual IRA arms decommissioning.
At present, most party activists appear minded to turn down the deal. But the more optimistic observers are pointing out that Ulster Unionists have a traditionally lengthy gestation period, and, during that period, opinion often moves from the extreme to a more moderate position. Some observers also believe that the fact the party did not make a knee-jerk rejection to the deal over the weekend is a promising sign it could eventually be accepted. One key moment will come this Friday when the party executive considers the proposals.
The party's 27 Northern Ireland Assembly members met yesterday to consider their reservations. Later, Dermot Nesbitt, a senior member of the UUP negotiating team, said: "At this moment we couldn't say yes. We're trying to see if we can clarify the position, and want to get to a position where we run into government and decommissioning. It's getting the conditions right and it's making sure the party is with us and we're with the party."
The result of having a multiplicity of spokesmen was illustrated by a noticeably less conciliatory tone struck by Jeffrey Donaldson MP. He said: "The Prime Minister is asking us, without certainty, to sit in the government of Northern Ireland with the representatives of a fully armed terrorist organisation. I find that unacceptable."
The strategy of not yet giving an answer means that in the run-up to 15 July Mr Trimble will be pressing Mr Blair for more concessions and a strengthening of the assurances he has already given. The Ulster Unionist leader is under heavy pressure to go along with a deal strongly supported by London, Dublin, Washington and most of the other parties.
Some in the Unionist party give the impression they are adamantly against the deal, even if new assurances can be extracted from the prime ministers. Many appear not to have definitely made up their minds.
The Assembly's Deputy First Minister, Seamus Mallon, added his voice to the calls for Unionists to accept the proposals drawn by Mr Blair and Mr Ahern. He said the outside world would not understand if Mr Trimble and his party rejected them.
"I do not give blank cheques," he said. "What I have given - and will give - is the guarantee that those who do not honour the commitment to decommission under the agreement will be removed from office."