It is hoped the invitation will build on a bid to bring the Pontiff to Glasgow in March next year.
He has been invited for a pastoral visit to the city on the 400th anniversary of the death of St John Ogilvie.
The Catholic Church is understood to have asked the leader of East Renfrewshire Council, Jim Fletcher, to request a blessing from the Bishop of Rome for the new school in Newton Mearns.
The proposed £13.6million joint faith campus will be a replacement for Jewish primary, Calderwood Lodge, and provide an additional Catholic primary alongside St Cadoc's.
In a letter to Archbishop Philip Tartaglia, Councillor Fletcher wrote: "In East Renfrewshire we have been working with both the Catholic and Jewish communities to see if there is support for a faith campus for a new Catholic primary school and for Calderwood Lodge.
"There has been support from both communities and although there are still many negotiations to be had, we are hopeful that a joint faith campus can be viable. We believe that such a faith campus may be the first in Europe.
"If Pope Francis was able to visit Scotland in March 2015, we wondered if it would be possible for him to support the joint campus perhaps by blessing the foundation stone for the school."
It is understood the blessing of the foundation stone would be done in Glasgow so that Pope Francis will not have to travel to Newton Mearns.
An East Renfrewshire Council insider said: "The Catholic Church contacted the council leader to ask for this letter. They're obviously trying to make a strong case for a Papal visit."
A spokesman for the Catholic Church said blessing the stone is a "fitting gesture" they would be happy to support if the Pope visits Glasgow.
He added that it would be for the Holy See to decide if the Pope is able to accept the invitation.