The incident happened on Carlton Place at around 6.30pm Thursday.
The victim, who is believed to be a lawyer at a city centre firm, was taken to Queen Elizabeth University Hospital with serious injuries to his face. It is understood a knife was used in the attack.
Officers cordoned off the area while investigations were carried out.
A Police Scotland spokesman said: “At around 6.30pm on Thursday, a 63-year-old man was seriously assaulted in Carlton Place, Glasgow.
“The injured man has been taken to the Queen Elizabeth University Hospital in Glasgow where he is being treated for facial injuries.
“Hospital staff describe his condition as stable.”