As police stopped another parade crossing the River Lagan due to fears of sectarian violence in the district, the leadership said it was prepared to have face-to-face negotiations with representatives of the Catholic community.
Tommy Cheevers, of the Ballynafeigh branch of the Walker Club of the Apprentice Boys of Derry, whose members were stopped by the RUC on the Ormeau Bridge yesterday, said: "I'm ready to meet with them directly."
Members of other loyalist orders have refused meetings with nationalist residents' groups because of alleged links with republicans.
Yesterday's parade, which signalled the start of the loyalist marching season, passed off without any trouble. Even though the Apprentice Boys claimed their offer had been turned down, their request for direct talks chaired by an independent mediator represents a significant shift.
Mr Cheevers, once a member of the Northern Ireland Parades Commission, said: "We are trying to bring about change in our country. We need more tolerance and a better understanding of each other.
"We are ready to address the concerns of these people, but here they are running away from dialogue."
Up to 3,000 parades are due to be held in Northern Ireland this year. Most will be trouble-free but there is increasing public alarm over the potential for sectarian violence, especially in Drumcree.
With the peace process already under pressure, politicians on all sides fear that failure to resolve the critical issue of disputed parades could have an enormous impact on the communities this summer.
Gerard Rice of the Lower Ormeau residents claimed the Apprentice Boys' offer of talks was not serious. He said: "The only way we are going to solve this is by a process of conflict resolution. We were hoping to start later this year, but it would seem from this stunt today that people are just out to wreck what process we were hoping to build.
"I will speak with the Apprentice Boys in five minutes, but talks alone will not resolve the marching issue. They would help, but when are people going to show respect for this community?
"We will not accept sectarian and triumphalist parades coming through."Reuse content