Ecclestone has repeatedly cast doubt on Silverstone's place on the Formula 1 schedule in recent years, and called on the circuit to substantially upgrade its facilities.
Proposals were recently submitted for an all-new pits complex, and Hill - president of circuit owner the British Racing Drivers' Club - is now confident that the revamp will be completed within the timeframe specified by Ecclestone's Formula One Management organisation.
"I am delighted to report that we are close to finalising our plans to build new pits in time for the 2010 British Grand Prix, thus satisfying FOM's pre-requisites to granting us a new F1 GP contract," Hill said.
The 1996 world champion said that he was hugely proud of the work being undertaken at the track
"Silverstone's new state-of-the-art facilities are therefore moving from aspiration towards reality," said Hill.
"The BRDC and Silverstone team has achieved an enormous amount over the last year.
"I am personally both impressed by and proud of the commitment shown by the BRDC and by all our team members engaged in this project.
"The BRDC share this desire and enthusiasm with thousands of British racing fans and we are doing everything within our power to keep the grand prix at the home of British motorsport."
But Hill added that he hoped Ecclestone's organisation would be 'reasonable' when it came to discussing a contract extension for Silverstone.
"The negotiations with FOM are ongoing and reasonable terms on a new contract will help us to fulfill our ambitions," he said.