The Italy international is said to be keen on joining the Bianconeri after a training-ground clash with Jose Fonte opened the Saints up to the forward's possible departure
The Bianconeri have confirmed their interest in the Italy international, who was suspended for two weeks earlier in January after an incident at the club's training ground that the Saints considered "a breach of the conduct expected of its players".
"At the moment there's no news regarding Dani Osvaldo," Pochettino told the press on Thursday, "but there's still a day-and-a-half left to finish the transfer activity, so anything can happen.
"I don't know because I'm not in charge of that situation [but] anything is possible.
"For me, the biggest talking point of the day is the game against Fulham [this weekend]; anything else is an individual matter that will be decided later. It's something that the club will evaluate if it does indeed happen."
The Argentine added that he does not expect the club to be involved in any other potential deals on deadline day, as he is satisfied with the options available to him and has no intention of selling Jack Cork, who has reportedly been the subject of a bid from Swansea City.
Pochettino said of the midfielder: "He is a very important player for us. There's no possibility of him leaving.
"I enjoy [deadline day] but I'm calm. Twenty-four hours is a long time in football. We'll be analyzing what other clubs are doing but we know that our squad is already good enough."