Reports have suggested Osvaldo, who only joined Saints in the summer, is unsettled at St Mary's and is eager to move on despite penning a four-year deal at the club.
Osvaldo made the switch from Roma to the south coast in August and became a Southampton club record signing, with the club paying £12.8million for the Argentinian forward.
However, Pochettino dismissed suggestions Osvaldo will leave the club, insisting he is in need of a confident boost as he continues to adapt to the Premier League.
"Nowadays with social media like Facebook and Twitter anybody can put any type of news out there and so we don't really pay attention to that," the Argentinian manager told the Daily Echo.
"We take those things for what they are and focus on the reality. The toughest thing is that on social media you can't prove anything or disprove anything.
"Anyone can throw any kind of news out there and it's hard to disprove or prove that news. For us the most important thing is that we know the personal situation of every single player and we just have to keep going and not pay attention to that."
Prior to his summer switch, Osvaldo had been plying his trade on the continent in Serie A and Pochettino believes the striker deserves time to familiarise himself with the English game.
"He's still in the process of adapting," Pochettino added. "There's a big difference between the Spanish, Italian and the Premier League.
"After getting to know the French league, the Italian league and the Spanish league I think being here in England this is the toughest competition.
"Every player needs their own personal time to adapt to this league. Every player is different but I am sure he won't have any problems adapting to this league and I'm sure that in a few months we are going to see the best of him."
Osvaldo has been named alongside fellow forward Steven Davies in the Saints squad for Saturday's trip to face Stoke and Pochettino is delighted to have the duo back after both missed last weekend's win at Fulham through injury.
"It's a good problem to have," he said. "It makes me very happy that both Osvaldo and Davis can return to the starting team and in the game against Fulham we played as a team and got the win and that's the most important factor.
"It just shows me that the 23-24 players that I have available can all do well for this team and then it's down to me as the manager to make the decision as to what players will be playing."