“The group is transitioning into an underground organization that places more weight on asymmetric tactics, like suicide bombings against soft targets in government-secured areas like Baghdad,” said Otso Iho, a senior analyst at Jane’s Terrorism and Insurgency Center at IHS Markit in London.

Mr. Iho cited an attack by two suicide bombers in Baghdad last month that killed three dozen people and injured 90 more. The attack took place in a busy Baghdad square where day laborers gather to look for work.

Estimates of how many fighters may have escaped into the deserts of Syria or Iraq and beyond are difficult to pin down, but American and other Western intelligence and counterterrorism analysts with access to classified assessments put the number in the low thousands. Many are traveling with spouses and children who are likely to have been radicalized during more than three years of Islamic State control of the region and could pose security risks as well, analysts say.

In December, Col. Ryan Dillon, the chief spokesman for the American-led military campaign in Iraq and Syria, said in a briefing with Pentagon reporters: “Syrian regime commanders in eastern Syria suggest that ISIS fighters” from the Middle Euphrates River Valley “may have slipped through porous Syrian and Russian defenses to arrive in areas near Damascus.”

Asked late last month by The New York Times about indications that as many as 1,000 fighters and family members had fled the Euphrates River area just in recent days, Colonel Dillon’s command replied in a statement: “We know that the Syrian regime has given ISIS the leeway to travel through their area of operations, but we cannot confirm any alleged incidents or operations that are taking place outside our area of operations.”

The United States military is concerned that a Turkish offensive against the Kurdish-dominated Syrian Democratic Forces in Afrin, in northern Syria, has worsened the problem. The S.D.F. has been working with the Americans in former Islamic State-held areas to interdict fleeing jihadists, but those efforts have been greatly reduced as the Kurds have shifted resources to reinforce Afrin.