The elders said should Buhari fail to seek re-election in 2019 or becomes incapacitated, the North must retain the presidency.
They said should anything worse happen to the ailing President Muhammadu Buhari, his deputy, Osinbajo cannot replace him in 2019.
Buhari, whose ailment is not disclosed, returned to London last week for a medical follow-up.
According to the Chairman of the Forum, Professor Ango Abdullahi, should Buhari fail to seek re-election in 2019, the North must retain the presidency.
He said the region would field a candidate through the ruling All Progressives Congress (APC).
He said although Osinbajo has the constitutional right to complete the current tenure being Buhari's Vice, he must not push his luck in 2019.
He said, "the constitutional provision says that in the event that the president dies in office or becomes incapacitated for one reason or the other and cannot perform the duties of his office, he will be succeeded by the Vice president.
"But no one is in a position to know at what point he is going to recover fully and come back to office, or whether the unexpected could happen. The northern elders insist that Osinbajo cannot replace Buhari in 2019.
"If the worst case scenario happens, and Buhari is not back on seat, the constitution is very clear that the vice president takes over.
"And what we then expect is that he will take that tenure to its logical end and from there, the politics of power sharing will take precedence over anything else."
Abdullahi added that "When the unexpected happens, in 2019, we will be back to our position that the north should be treated equitably and fairly.
"And we expect that another northern candidate will emerge as presidential candidate on the platform of APC."