Footage of Brown leaving Ibrox went viral on social media after a voice is heard asking the player, "How's your sister?"
Brown's sister Fiona died aged just 21 from skin cancer in 2001. In the video, Brown can be seen staring at the supporter before boarding the Celtic team bus.
A 15-year-old boy was yesterday charged in connection with the incident and banned from Ibrox for life over the vile taunt.
The Daily Record is reporting that Robertson personally contacted Brown to convey his "disgust and regret" at the flashpoint. The Gers chief is understood to have phoned Brown, and apologised in an email to the Celtic captain.
Brown is said to have been "heartened" by Robertson's response along with the messages of support he has received from both sets of fans while Celtic chiefs are believed to be satisfied with how Rangers have responded to the incident.
A Celtic source told the Record: "Scott was obviously deeply upset and angry about what was said but he has taken a great deal of comfort from the support he has received from all across the board over the last 24 hours.
“He didn’t want an issue to be made of this but, now that it has become public knowledge, he is grateful for support he’s received from all over and the swift and sincere response he’s had from Rangers as a club.”