Brown limped off early in Sunderland's 1-0 defeat to West Ham on March 21, with initial tests indicating that the defender could be out for the rest of the campaign.
However, Advocaat feels the former Manchester United man could be fit for the final weeks of the season.
"It looks better than we thought before," the Dutchman told Sunderland's official website. "It will still be difficult to bring him back this season, so hopefully at the end but the doctor said it takes four or five weeks."
Brown will miss Sunday's Tyne-Wear derby, but Lee Cattermole is available again to Advocaat following his two-game suspension.
"You need him [Cattermole] always because he is a controller on the pitch and he is a quick passer," Advocaat added.
"The team missed him for 10-12 games I believe so it is good that he's back."
‘He instantly popped into our ratings as one of the best U21 midfielders in Europe. The impressive thing is Slot has made him into more of a no.6' Inside Ryan Gravenberch's transformation at Liverpool
‘I don’t think Liverpool would look at Ollie Watkins, a striker isn’t a pressing issue for them – it’s Arsenal who need one’ Former Reds star explains why his old club don’t need an out-and-out forward this summer
