The Italy international has done little to justify the club-record fee paid to Roma in the summer and, worse still, has irreparably fallen out with many Saints players.
Inter Milan, Sevilla and West Ham have been mentioned as potential destinations, although it is Juventus that appear to be leading the race for the 28-year-old's signature.
Juve general director Giuseppe Marotta has confirmed their interest in the forward but Pochettino insisted no deal had been struck.
"At the moment there is no news regarding Dani Osvaldo," he said.
"There is still a day and half left to finish the transfer window activity so there's many things that can happen in that time.
"For me the biggest talking point of the day is the game against Fulham, nothing else.
"Anything else is an individual matter that will be decided upon when it has to be decided upon."
Pochettino said he was "not in charge of that situation" but was able to confirm that Saints had turned down a bid from Swansea for Jack Cork.
The former Argentina international also suggested the need to bring in a new face has lessened somewhat thanks to Sam Gallagher's impressive senior debut in midweek.
Despite failing to score, the 18-year-old led the line with aplomb against Arsenal - a performance that has seemingly earned Pochettino's trust.
"It's clear that on Tuesday Sam Gallagher played and played very well against Arsenal, a product of our academy and our team so why not give him the opportunity to keep playing?" he said.
"I am not saying that [we will not sign anyone] but signing right now someone who fits the profile we are looking for is not easy. It's very complicated to find that player.
"With Sam Gallagher's performance against Arsenal on Tuesday it made us confident we might give him a chance and keep counting on him because he showed he can compete at the top flight."
Another youngster set to get a chance with the first team is teenage full-back Matt Targett.
The England Under-19s international looks set to be bumped up to Luke Shaw's deputy at left-back after Danny Fox was allowed to join Nottingham Forest on loan, with a permanent move agreed for the summer.
"The first thing I want to make public is my gratitude and thanks to Danny Fox for his professionalism and attitude during my time here at Southampton," Pochettino said.
"We want to wish him the best because he's shown great character and great qualities as a player and a person. We wish him all the best at Nottingham Forest and his new challenge there.
"Matt Targett is training with us and he has a chance to get into the side if we see that he's able to perform at the level that we expect."
Fox is one of a number of squad players to have left Saints this month, following Lee Barnard, Billy Sharp, Tadanari Lee and Aaron Martin out of the door.
The latter joined Birmingham until the end of the season on Thursday, while Jason Puncheon is understood to be close to making his loan move to Crystal Palace permanent.