The groups said the death toll during Africa’s biggest election were higher than that of the 2015 poll which was widely considered to have been orderly, aside from a Boko Haram attack that killed more than a dozen people.
Past polls have been marred by violence and vote rigging.
Clement Nwankwo, convener of the Situation Room, which represents more than 70 civil society groups, said 16 people were killed in election violence across eight states, while Lagos-based consultancy SBM Intelligence cited 35 deaths.
Police did not immediately respond to phone calls and text messages seeking comment.
Buhari, 76, a former military ruler who is seeking a second elected term at the helm of Africa’s top oil producer on an anti-corruption platform, faces Atiku, 72, a former vice president who has promised to expand the role of the private sector.
Festus Okoye, an Independent National Electoral Commission official, said some people were killed in the southern states of Rivers, Lagos and Oyo, but he could not confirm any figures.
A handful of polling stations remained open on Sunday to allow more Nigerians to vote, after the violence, delays in opening poll centers and malfunctioning systems in various states interrupted voting the previous day.