Mr Osborne said a concert hall could give “significant artistic, educational and economic benefits” to London, while Mr Johnson declared that it would cement the capital as “a world city for culture”.
Some hope that a new hall will tempt Sir Simon to return to Britain and take over the London Symphony Orchestra.
“London is a world-class city with many fantastic cultural assets,” Mr Osborne told the Evening Standard.
“I want to make it even better, so I am delighted we are looking at this with the Mayor as part of our long-term economic plan for London.”
He added: “Speaking to the likes of Sir Simon Rattle has impressed on me the significant artistic, educational and economic benefits that a modern concert hall would bring not just London but the whole country.”
The concert hall would also be cutting-edge digital and educational facilities to spread the benefits and the venue would have to complement rather than compete with London’s existing facilities.
Mr Johnson said: “We have heard the clarion call from Sir Simon Rattle and many others who wish to see a brand-new and world-class centre for music in London.
“The feasibility study being confirmed today will enable us to understand fully the potential to build that centre and help to cement our position as a world city for culture.”