Demonmir approached Nina intending to kill her.
Ottoman studied Eren's background report carefully. He always liked to find his enemies' weak spots after all. All so that he could use them and manipulate the rankers however he wanted.
After studying him a bit, Demonmir knew Eren was an oddball. And that he didn't show an in-depth emotional investment in anything. Demonmir concluded that his vulnerability was Nina.
After all, why would Eren buy Healers' Nirvana and attract Midnight Moon's attention for himself? Especially when the butcher's profile made it very clear to Demonmir that he liked to avoid unnecessary troubles.
Demonmir's spies also told him about Carvo Bianco Eren made for Nina. That place was cared for by the butcher himself.
These seemingly insignificant details didn't matter much when looked at separately. But in the end, they created a distinct picture for Demonmir. He concluded that Eren's actions could be manipulated by using Nina as a chess piece.