Investigators didn't title the person however police chatting with AFP recognized him as Sokolov.
Some students said he was a "freak" who called his lover "Josephine" (Napoleon's first wife) and asked to be addressed as "Sire".
They had also co-authored a number of historical papers together at St Petersburg State University, where Sokolov was a professor in the history department.
"He has admitted his guilt", Sokolov's lawyer Alexander Pochuev told the outlet, adding that his client was cooperating with authorities.
He had successfully thrown her hands and legs into St Petersburg's Moika river, according to the newspaper, but slipped and fell into the ice-cold waters on his third visit.
A former student also claimed that Sokolov was dedicated to his work but was also emotionally unstable and abused alcohol.
A court on Monday will decide whether to arrest the historian, who was being treated for hypothermia in a hospital.
The relationship was said to have been an "open secret".
Search and rescue workers in the Moika River.
A distinguished Russian professor chopped up his student-turned-lover and planned on killing himself while dressed as Napoleon Bonaparte - but was instead busted carrying a backpack stuffed with the woman's arms, after drunkenly falling into a river, according to reports.
A historical re-enactor, Mr Sokolov has often appeared in uniforms of the Napoleonic era, and has worked as a consultant on historical reproductions for film and TV.
Sokolov was president of the Russian Association of Military History and is now a member of the body's scientific council.
Both he and Yeschenko enjoyed wearing period costumes, with Sokolov frequently dressing up as Napoleon.
Sokolov, 63, who received France's Legion d'Honneur in 2003, was arrested on November 10, 2019 on suspicion of murder after he was hauled out of Saint Petersburg's Moika River with a backpack containing a woman's arms.
Sokolov was also a member of the Institute of Social, Economic and Political Sciences (ISSEP) in Lyons, France, which has been described as a far-right academy.
According to police statements, Sokolov said he killed Yeshchenko after the pair had got into a fight.