The use of auctions in procurement results in price reduction as well as the reduction in the cost and the time required to complete transactions. In many situations, price-only auctions resulted in the violation of the contracts or even contractors' bankruptcies. This is one reason for the introduction of multi-attribute auctions as well as auctions-followed-by negotiations. Auction theory is based on two assumptions which assure that auctions are efficient mechanisms, guarantee efficient solutions, and produce the best possible results for bid-takers. In practice these assumptions are often violated. The contribution of this paper is to propose a procedure for auctions-followed-by negotiations which retains important auction features such as process transparency and efficiency while allowing for increased social welfare. The unique feature of the procedure is the introduction of the win-win phase in which the market participants may attempt to make joint improvements to efficient solutions obtained from auctions.