While what the other attorney stated is true in federal court, in Arizona state court, both Plaintiffs and Defendants make Offers of Judgment under the rule. The idea is to set the amount at a level you believe you can do better than if the matter goes to trial. They are especially useful in settling medical malpractice cases because of the great expense of expert witnesses. If the defendant refuses to accept the Offer of Judgment and does not do better at trial, s/he is on the hook not only for the judgment but also for double taxable costs AND all of the Plaintiff's expert witness fees. So the amount should be set not based on the Plaintiff's bottom number for settlement but on what you realistically think you could recover at trial.
An offer of judgment is made under a specific rule of civil procedure intended to encourage settlement. In federal practice, it is Rule 68. A defendant usually sends an offer of judgment to the plaintiff. If the offer is not accepted, and the plaintiff recovers less at trial, then the plaintiff is liable for the costs after the offer was made (see FRCP 68). While a plaintiff is not precluded from making an offer of judgment, it is more common for the defendant to make such an offer under the rules of civil procedure. A plaintiff usually just makes a settlement offer without the formalities of a Rule 68 offer of judgment.
Your question, however, seems more focused on the art of negotiating, rather than on the formal rules of civil procedure. There are many different styles of negotiating (cooperative, confrontational, etc.), and one may adopt a different negotiation strategy for different situations. But, as practical matter, a plaintiff would have difficulty getting a defendant to accept a settlement figure higher than its initial offer unless there is a change in circumstance to justify it, A plaintiff could say that it will settle for a lesser amount today, but a higher amount later if it is forced to litigate the matter. But a defendant may interpret that as bluster and ignore the threat. A lot comes down to how well you communicate and understand the other side.
This answer is for informational purposes only. It is not intended as specific legal advice regarding your question. The answer could be different if all of the facts were known. This answer does not establish an attorney-client relationship.