Do project managers negotiate contracts with clients?
It looks pretty reasonable that a person who manages the project should be involved in this project right from the start and clearly understands all the conditions of doing business with a client, especially the payment terms of the contract/project. And not only should they know these conditions, but they should negotiate and demand these conditions and terms be realistic (from the technical point of view) and mutually advantageous (from a financial point of view). But, at least in my experience, project managers are often not involved in a potential project from its beginning and don't have access to financial/payment terms.
In your experience, do project managers negotiate contracts with clients?
Contracts are Legal Documents, and Project Managers Aren't Usually Officers of the Company
This question is likely to be closed as too subjective, but a lot depends on the size of the company. In most companies beyond the mom-and-pop size, contract negotiations are usually the domain of the sales team and the legal department because they are binding agreements on the company. Binding the company is usually outside the scope of the role or profession of project management, although there are always edge cases.
Even in smaller companies, most project managers are neither legal experts nor company officers, and are therefore usually neither empowered to legally bind the company nor well-suited to opine on the specific legal terms of the contract. However, if you have standardized contracts where the scope, budget, or schedule are simply variables, then I would certainly hope that the sales team, legal team, and company leadership would at least consult the folks expected to do the work (including the people managing project delivery such as an Engagement Manager and/or Project Manager) to ensure that the work is likely to fit within the negotiated terms.
That said, in my personal experience this rarely happens. Senior leadership often sets sales and market-related targets to fit current business goals whether or not they are reasonable; salespeople often makes sales to meet quotas or commission goals regardless of whether the development or delivery teams can deliver what's promised; everyone else just tries to get the scope done within the budget and schedule provided.
No one is saying it should work that way. However, if your question is simply whether you (presumably as a project manager) should be directly negotiating with clients, the answer is most likely "no." You should certainly be partnering with sales, legal, leadership, and your team(s) to ensure that project initiation goes smoothly, and to identify any red flags or change orders in scope, schedule, or budget. Whether or not doing these things fits your company culture or management's expectations is a question for your senior leadership.