The BRDC, which operates the track, has frequently been threatened by Ecclestone to have the rights to stage the race withdrawn, amid concerns over the facilities.
But Ecclestone says he believes he will be able to talk business with the former F1 World Champion to secure his long-term role.
Said Ecclestone: “They have got a new board of people now so maybe they will do something. I can deal with Damon easy – we'll see what happens.
“We want to keep the British Grand Prix and we'll see if we can make it happen."
The current deal to host the event expires in 2009.