WASHINGTON — Buoyed by support from its NATO allies, Turkey escalated its warnings against Syria on Tuesday, even as some US and allied officials privately raised questions about whether the Turkish warplane shot down by Syrian air defenses — provoking the denunciation — had been on a spy mission.

In response to the downing, Prime Minister Recep Tayyip Erdogan of Turkey warned Syrian forces to stay clear of their mutual border or face a military response to any perceived threat. Erdogan’s bellicose tone came as ambassadors from the NATO alliance, seeking to avoid a wider conflict, held emergency talks in Brussels at Turkey’s behest.