The England international has not played since the end of January when he damaged knee ligaments during a goalless draw with Sunderland at the Stadium of Light.
Yesterday, the North London outfit revealed that Rose has undergone "exploratory surgery", and it has been reported that the left-back could be on the sidelines until September.
"We need to see how he reacts to the situation and then it is taken day by day to see how he recovers," Sky Sports News quotes Pochettino as saying. "We hope and wish that he has a speedy recovery because it will be good for us and good for the team.
"It's a difficult moment for him, but we're here to support him and to help him to be available as soon as possible. It's a period in his recovery that we cannot accelerate and we cannot push too much. It's how his knee reacts.
"I sent him a text after the surgery and we crossed texts, but it's a moment to give time to the recovery after yesterday. He'll be here on Friday, to see and check how he is. Of course, we have a great medical staff and they will help. He has to be good and strong for when it will be possible [for him to return]."
Rose will miss England's internationals against Scotland and Germany this summer.