Fist of all, I HATE negotiating. It always makes me so uncomfortable and awkward. We're just getting our bids back and going through them. How do you go about negotiating with yours to get them lowered?
The purpose of bidding is to avoid negotiating. The lowest qualified bid should be awarded the contract. However, if the prices are all above your budget you can ask all of the bidders to submit suggestions to reduce the cost of the project and then bid the revised design again. Or you can ask the lowest bidder to sit down and negotiate a price with you but negotiating with all of the bidders after the bids have been opened is contrary to the purpose and spirit of fair bidding.
While Renovator8's response is entirely valid and correct, I have personally broken the spirit of fair bidding more than once, and been quite happy with the results.
On several projects (in our current home, not the one we are building) I have received multiple bids and in cases where the lowest bid wasn't my first preference I have gone back to the contractor I preferred to work with and asked him if he could match or beat the price. Once the contractor couldn't, and twice they did. It can never hurt to ask. Seperatly, on one project (granite) I just told the contractor what my budget was and he made it work, even though it was about 15% less than his bid.
That said, you have to have an idea of what a fair and reasonable price is before negotiating.
If all of the bids are coming in over your budget, then your budget isn't realistic. That's not a place to "negotiate". That's a starting point for you to start cutting out features from the home.
Asking a bidder to match the bid of another bidder is not "negotiating"; it is "price shopping" and entirely contrary to the generally accepted standards of competitive bidding. Sure, you can probably get away with it but I doubt any of the other bidders would be willing to bid another of your projects if they knew you would reveal any part of their confidential bid submittal to another contractor in order to pressure him to lower his price.
If there is no award after the bids have been received, it is OK to inform all bidders of the bid amounts or to ask all bidders to make alternate proposals or offer suggestions for lowering the project cost but it is not acceptable to privately ask one bidder to beat the price of another bidder. Bidding is not only a way for an owner to possibly lower the project cost, it is a way to allow all selected bidders to have a fair shot of getting the job.
Advice we had, which seemed to work well and didn't cause any apparent difficulties was to ask contractors if they would like to reconsider particular line items in their bids (without revealing any information, or even whether we thought they were high or low). We did this both where we suspected errors because the numbers were much lower than expected and where a preferred bidder was a high outlier in a particular area. Generally the bids on each line item were within 5-10% of each other, so when one bidder came in with concrete work at 50% more than the others we asked him to look at that number again and he came back with something still slightly high, but within the expected range.
If all the bids are high the appropriate question is how to reduce the scope of the project to something that you can afford. We spent a hard couple of months paring a foot here and there to reduce a wonderful 1400 sf addition to a 1050 sf addition we could afford and still like.
In our case we got bids with a preliminary plan and one guy was a clear winner. After a "hard" bid with the final plans and site plan he became the highest. We went with the lower guy since he knew the project best for a variety of reasons. However when we let the middle contractor that we were going to use someone else. He asked us if he can look over his numbers again and see where he may have missed it. He came back 30k cheaper and almost as low as the other guy. He had talked to a few subs and found some free dirt (we needed a massive amount).
We never told him the other guy's numbers, but he was really hungry for the business so cut out some of his profit too. In the end we still chose the lowest guy, but I was impressed with the other guy too.