Osinbajo also disclosed that the Redeemed Christian Church of God was initially reluctant to release him.
Osinbajo, however, did not mention the person's name but described him as a former governor - the description is believed to be Bola Tinubu.
Buhari's recently-launched biography, written by Professor John Paden, had sparked controversy over who nominated Osinbajo as Vice President.
Before the book, it was generally believed that Osinbajo was chosen by Tinubu.
But Paden claimed in the biography that Tinubu actually wanted to become the Vice President but was handicapped by the religion factor - a Muslim president and vice president ticket would have automatically won the race for the then incumbent President, Goodluck Jonathan.
The book claimed that when it was apparent that Tinubu couldn't run, he opposed the idea of Osinbajo's nomination when it was raised by one of the party's chieftains.
Speaking at a State of the Nation Address convened by Coalition of Nigerian Apostolic Leaders in Lagos, Osinbajo said he was put forward by a leader in the APC and accepted by other leaders.
He said: "An APC National Leader who had served when he was governor was interested in putting me forward as the running mate of President Muhammadu Buhari in consensus with other national leaders of the party.
"I was a product of this system (Coalition Leaders). The church was reluctant and I understood why the church was reluctant.
"The church was looking at the head of the ticket (President). This is our first opportunity. This is our first shot at it. We need to maximize it and do the best we can. I think what my nomination and appointment have done is that it has open the door for us. We can do it and we can be influential in doing it."
Tinubu had also dismissed the account contained in Buhari's biography.