The cost to demolish a driveway is usually somewhere between $2 and $6 per square foot. So you could reasonably expect to save somewhere between $2400 and $14400 off the contractors bill. Possibly more if your location is particularly annoying or if the contract you end up going with has a "I really hate demo work" cost built into their quote.

There's a lot of factors that go into it, hence the hugely widely variable cost.

Really the only way to be sure is to get a quote with and without demo.