I would be surprised if they do. I think a relationship partner in a large law firm may let a general counsel of a primary client know somewhat beforehand that the merger is in the works or about to be consummated, but not seek approval. The conflict of interest research and determinations by the firms should suffice.
Early disclosure, even to a trusted client, presents some risk of a leak. It might also be hard to decide which clients to approach and how to characterize the merger’s effect on service to them. An experienced general counsel might have some observations about the reputation of the merger partner or the advantages – or disadvantages – of firm size and geographic scope, but one would be hard-pressed to imagine how comments on those topics by a single client would derail or clinch a merger that otherwise makes sense to the two firms.