President Blaise Compaore resigned at the end of October after more than two decades in power.
Violence broke out in the streets as he tried to extend his reign and stop democratic elections.
The military in the country took over and then asked for the church to provide a 'President of the transition'.
It's reported the offer was aimed at Cardinal Philippe Ouédraogo of Ouagadougou.
But the Church says this was not possible under Canon Law and that a clergyman cannot take a political position.
In a statement spokesman Fr Joseph Kinda said: "The Cardinal wants to tell us that by choosing to imitate Christ in the preferential option for the poor, a priest must avoid creating new forms of exclusion by being too partisan.
"The risk is enormous, as demonstrated by the experience lived by some local churches. So in addition to the canonical regulation, the position of the Cardinal is eminently pragmatic."