The Defense Department plans to send a few dozen U.S. troops to the Middle East in the coming days to be ready to help Americans flee a region bracing for a significant escalation in the simmering conflict between Israel and the Iranian-backed militant group Hezbollah in Lebanon, according to three defense officials.

The U.S. forces will add to the roughly 40,000 troops already in the region — a presence that has grown as the war between Israel and Hamas continued and tensions with…