Jeffrey Sharkey, who took over the role this week, said: "Things we are considering, if we have the money, I would love to build out on the front of this building.
"[It would be] a lovely front of house space, a more welcoming presence to the community."
The Royal Conservatoire has occupied its current purpose-built brick building on Renfrew Street in Glasgow since 1988, and it also has buildings at Speirs Lock.
He added: "We put on so many concerts and productions and recitals and plays in our five halls here, but the way the building is designed you have to come up all these stairs and a lot of the public may not know those are going on, despite all our efforts of publicity.
"They may think: 'Well that's where you go to study piano', but in fact inside there's a vibrant play going on. Having something that shows our wares, extends our space, makes it look a little more glamorous."
Mr Sharkey said the The Juilliard School in New York was a good example of a building that had made itself more appealing to the public.
The Brutalist building underwent a major renovation in 2009.
"Before it was even more of a granite block than we are - although we are not granite - they were a white block and you couldn't easily find the entrance: but that new thrust it has, all lit-up at night, it shows that this is a place where the arts happen.
"That's a statement I would love to dream of making."