Mr Ashkenazy had threatened to resign earlier this week after he learnt that the orchestra was secretly discussing his replacement.
In a statement made yesterday, the orchestra's board said it wished "to apologise for the hurt that has been done to their music director, Vladimir Ashkenazy, and recognise that it was their duty to have consulted with him about any possible successor".
It is understood that the board had been talking to the conductor Daniele Gatti, one of the classical music world's brightest young stars, who is already the principal guest conductor at the Royal Opera. Mr Ashkenazy, 57, best known as a pianist, joined the orchestra more than 10 years ago. His contract has a year to run.
The orchestra's statement continued: "The RPO further acknowledges Mr Ashkenazy's long and devoted commitment to its interest and his generous financial support offered through fund-raising concerts performer without fees, reduced conducting fees and other irreplaceable contributions."
The statement added: "Further discussions about his future work with the RPO will take place in due course."
Mr Ashkenazy's agent, Jasper Parrott, said last night that the conductor would be working with the orchestra during the UN World Tour which is scheduled for June 1995. It marks the UN's 50th anniversary.