Plans were scrapped for a purpose-built home for the orchestra when the price tag came in at £30m.
Under the new scaled down plan, the existing Glasgow Royal Concert Hall will be refurbished and extended over its goods yard area.
Carillion beat Graham Construction and Gilbert Ash with a £14.1m bid.
The new 600-seat auditorium and rehearsal space will be designed to also provide an additional performance venue of international quality to complement the existing Concert Hall programme.
Leader of Glasgow City Council Gordon Matheson: “We’re delighted to be investing in one of Scotland’s premier performance spaces, which will create a centre of musical excellence for both the city and nation.
“These ambitious plans will help to realise new ambitions for the RSNO and enhance the Glasgow Royal Concert Hall as a world-class performance space. The physical refurbishment will also offer Glasgow Music greater flexibility in attracting both new and existing talent to the performance space.”