Short answer...Yes.
If they award the contract to the selected contractor under the premise that the project is over-budget and that changes will be necessary to get the cost down (I hate the term "value engineering" since it doesn't usually involve engineering and there's rarely any value consideration made other than to reduce costs), then they can negotiate with the contractor. Here's where the design "rub" takes place. Suppose through their cost cutting process they decide to do something that you as the design professional do not agree with. You will have to make it clear to them that you will not sign/seal permit documents for such. They will get all pissed off at you and possibly terminate your contract. I hope you have good terms and conditions in your contract that allow you to maintain control of your drawings.....otherwise they'll use them and likely get some other design professional to accept them as his/her own (not legal in most jurisdictions in the US, but it happens).
The public agency was not very bright to put a hard budget on an engineering estimate. We do not use the term engineering estimate in our work. The reason being is that "estimate" implies a greater accuracy (like a bid) than the engineer intends. We use the term "engineering opinion of cost", since opinions cannot be held to the same accuracy standard as "estimates".
Good luck.