Unionists have been asked to reconsider their refusal to join a body set up under the 1985 Anglo-Irish Agreement.
The body fosters links between the Irish and British parliaments
The DUP and the Ulster Unionists have also been asked to submit proposals to rebrand the British Irish Interparliamentary Body.
The Irish parliamentarian who chairs the body, Niall Blaney TD, said it was "incomplete" without unionists.
He has been in Belfast this week meeting the DUP's Peter Robinson and Ulster Unionist leader Sir Reg Empey.
Both unionists parties have refused to take part in the body since its inception in 1990 although Peter Robinson ended his party's boycott in 2006 when he agreed to address it in Killarney.
The 50-strong body comprises TDs, MPs and representatives from Stormont, Edinbugh, Cardiff, the Channel Islands and the Isle of Man.
Mr Blaney said the body needs unionists on board to be more effective.
He has invited proposals from unionists on a name change and a significant change in direction away from Northern Ireland politics.
He has suggested a new body could focus on tourism, and shadow the British-Irish Council's work on issues such as drug abuse.