Cases can settle at any time. While most cases settle before trial, nothing prevents litigants from negotiating after a case is appealed. I’ve had a couple of cases settle after an appeal is filed. So long as the ultimate result is uncertain–i.e., remittitur hasn’t issued–both parties have some incentive to reach an agreement. A procedural consideration for such settlements is which court has authority to approve the settlement: the court that issued the order being appealed or the court which has the appeal under consideration.
A May 20, 2013 order from the South Carolina Supreme Court in the case of Jim Lancaster v. Georgia-Pacific Corporation spells out the procedure for obtaining approval of post-appeal agreements. “[A] settlement may not be taken by the lower court, except with regard to matters not affected by the appeal, while the matter is pending before this Court. The parties must first seek to have the matter remanded to the lower court.”
Thus to get agreements approved that resolve an issue on appeal, one must either ask the appellate court to approve the agreement or must seek to have the matter remanded to the lower court for approval of the agreement.