The Nation said while police confirmed 10 women and one man had died, the death toll could go higher, with at least 25 other people injured, some seriously.
Four other people -- three security personnel and a civilian -- died in a traffic accident involving a convoy carrying dignitaries.
The newspaper said the stampede happened shortly after President Goodluck Jonathan began delivering a speech at a campaign rally of the ruling Peoples Democratic Party at Liberation Stadium in Port Harcourt, the Rivers state capital. The mayhem occurred as some people tried to work their way out of the stadium as others tried to get in, the newspaper said.
Stadium security personnel shut larger gates, leaving only one small gate where people attempted to force their way through, The Nation said.
Vanguard newspaper reported without attribution that the chaos was fueled by police on horseback riding through the throngs, who were trying to reach money tossed into the air by an unidentified political official.
The Nigerian Tribune reported a spokesman for Jonathan, Ima Niboro, said the president had ordered an investigation into the deadly incident.
"I am sad, and heavily weighed down by this incident," the president said in a statement. "It is sad, unfortunate and regrettable. I mourn with those who mourn tonight. May God grant us all the fortitude to bear this irreparable loss."