Side by side at the Executive
The Orange Order has said it has more than 300 reasons not to invite Martin McGuinness to a Twelfth parade, after the Deputy First Minister indicated that he would be prepared to attend.
The Sinn Fein MLA challenged the Order to invite him to one of the parades this July.
He also said he wants to "build a friendship" with new DUP First Minister, Arlene Foster.
A spokesman for the institution pointed out that all its demonstrations are public and anyone is free to attend them, but said it will not be issuing any invitations to Mr McGuinness and accused him of a political stunt.
He said the Orange Order has more than 300 reasons not to invite him to an event - a reference to its 333 members murdered in the Troubles.
Mr McGuinness told the Impartial Reporter that he believes the Orange Order should be "big enough" to invite him to mark the anniversary of King William of Orange's victory at the Battle of the Boyne in 1690.
"I think it's important not to impose yourself on, for example, the Twelfth of July, but I do think what is important is that people are big enough to invite you," he said.
"I would give it very serious consideration. I appreciate what the Twelfth of July means to the unionist community."
Despite having criticised some Orange demonstrations in the past, Mr McGuinness said he believes the presence of the Apprentice Boys in his home city of Londonderry "enrich our lives".
He said he takes "great pride" in the city's new Siege Museum when he walks past it with his dog Buttons once a week, but did not say if he had visited it.
Mr McGuinness said he would build a friendship with Mrs Foster, who has replaced Peter Robinson as DUP leader and First Minister.
"We have had a number of conversations and it's quite obvious to me that Arlene believes, and I believe, that we need to work positively and constructively together to ensure we put to bed the negativity of the last two and a half years," he told the Fermanagh MLA's local newspaper.
"The fact that she and I privately to ourselves have committed to working in a positive spirit is something to be welcomed.
"I can build friendships; I did it with Ian Paisley and Peter Robinson. I'll do it with Arlene Foster."
Responding to his comments, a spokesman for the Orange Order said Sinn Fein has never apologised nor shown genuine remorse for the hundreds killed by the IRA during the Troubles.
"There were 333 of our brethren murdered during the Troubles, the overwhelming majority by the Provisional IRA, of which Martin McGuinness was a prominent member and proud to belong," he said.
"Many of our murdered brethren were serving or retired members of the Crown Forces. Sinn Fein have never apologised, nor shown genuine remorse, for the murder of the Crown forces.
"To offer an invitation to Martin McGuinness would insult their memory and bring further hurt and distress to many families."
"There is also a certain irony in Mr McGuinness' remarks, given the organised and deliberate role taken by the republican movement in their continuing opposition to our parades," he said.
"This remains very much evident, with leading Sinn Fein representatives continuing to actively oppose and protest against our processions.
"The Orange Institution is actively involved in bridge building at many levels, but there are some things that as an organisation and as a unionist community are at present beyond the pale.
"We would encourage Martin McGuinness to continue on the political path on which he is now embarked, sadly his conversion from violence was too late for many, both Protestants and Roman Catholics.
"We would view his comments as 'gesture politics' rather than constructive peace building."