The 600-seat music venue, purpose-built rehearsal space and recording facility will be built on a site next to Glasgow's Royal Concert Hall.
The RSNO has managed to raise the £2m needed before construction could begin.
It has now launched a public appeal to raise the £500,000 it still has to find to pay for the £1m fit-out of the new building.
Orchestra chief executive Michael Elliott, said: "The RSNO has met its financial commitment to begin construction of its new home.
"Together with increased investment from the Scottish Government and Glasgow City Council and welcome contribution from Creative Scotland, the project team will meet the complex challenges of the build to provide world class music facilities.
"The outcome will deliver a bespoke rehearsal and recording space, exceptional education and learning facilities and a recital venue providing a valuable addition to the country's cultural venues.
"I am delighted we have raised all the funds required for building construction to commence and have raised half of the £1m required to fit out our new home."
Scottish Culture Secretary Fiona Hyslop said: "The building will enable the RSNO to share their new facilities with a range of other young musical talent, while the people of Glasgow will get a purpose-built music venue to further enhance their city's reputation as Unesco City of Music."