There's no evbidence about these Patriacrchs and their relations with freemassonry. The rumours, though, exist because of the close relations between the Church and the governments of Turkey and Greece, which after the desintegration of the Ottoman Empire have been fully controled by freemassonry.
This could have started before. At the time of Kyril Lukaris, this Patriarch had a very close friendship with the protestant and freemasson, George Mayer, who was in charge of "religious affairs" and had some interventions in the Church, meaning to "protestantize it". A new cathecism with calvinist tendences appeared, but it was never proved that the Patriarch had issued that cathechism.
It is clear that Christianity and Freemassonry are antithetical. Freemassonry has been condemned by the Orthodox Church, though I would wish a more fierce condemnation of it. Even the OCA has said this:http://aggreen.net/guidelines/guide05.html#Fraternal