In my experience the price is very, very variable, but it depends on the quantity and technology and where the work will be done in fairly predicable ways. It might be several US dollars per part for a small high-quality run of a few boards done in a first world semi-clean room environment to a tiny fraction of a penny per part for boards made to a consumer standard in a higher volume environment.
You need to find production houses that deal in your kind of requirements and will value your business, including volume. The 'A' list of production houses are probably not an option unless you're building in huge volumes and have a history and balance sheet to match, so you'll be dealing with a different set of suppliers. If they're used to dealing with (say) medical instrument companies they'll have a different set of systems in-house and different expectations for you, the customer, than a place oriented towards industrial or aerospace or consumer assembly. You have to shop around enough to make sure you're not overpaying, but also build relationships with a couple good suppliers so when things go pear-shaped you will get support. Hint: The lowest price supplier is rarely the best deal- more usually it's close to the middle of the pack.
In some cases it may be better to permit the assembly house to source certain parts, and that adds another level of complexity.
It sounds like you have boards already and can get quotes for assembly with minimal effort by simply e-mailing off the documentation. One trick for getting rough quotes if you don't have (or don't want to disclose) that information is to pick a similar board (in terms of complexity and technology) and get quotes on that, using that step to filter out the companies you don't want to try to establish a relationship with.