ISIS fighters have been fleeing from Syria into Iraq, perhaps with millions of dollars in tow

ISIS fighters have continued to flee even as the final fighting has unfolded in the group's last stronghold in southeastern Syria. Some of the last fighters are also believed to be former members of al Qaeda in Iraq, according to a second official.

The assessments and estimates of ISIS' strength come in the finals days of the physical caliphate, the officials said.

Earlier this month, Gen. Joseph Votel, the four-star general in charge of US military operations in the Middle East, estimated there were 20,000 to 30,000 ISIS fighters remaining -- which lines up with a UN estimate from August. A US Defense Department report from the summer estimated there are between 15,500 and 17,100 ISIS militants in Iraq and another 14,000 in Syria.

Contrary to US President Donald Trump's tweets, a senior US diplomatic official sounded a different and more nuanced definition of "defeat" when it comes to the ISIS caliphate.

Defeat doesn't entail only the physical caliphate, the official said when asked what defeat might mean. "We mean defeat of networks" of ISIS, including ongoing sources of additional financial revenue, those who provide them weapons and people who provide them with places to hide, the official said.

There may be tens of thousands involved in the effort, something the intelligence community has alluded to in congressional testimony.

Separately, in just one indication of Iran's growing influence inside Iraq, Brig. Gen. Qasem Soleimani, the head of Iran's Islamic Revolutionary Guards Quds Force, has traveled to Iraq as many as 20 times in the last three to four years, according to a senior US diplomatic official.

The US has not asked for Soleimani to be arrested but has brought to the Iraqi government's attention his apparent freedom of movement.

Soleimani is on Treasury Department and UN Security Council watch lists for those allegedly involved in terrorism.

The official described "aggressive" instances of Iranian militias intimidating Iraqi towns and villages, and moving into specific areas to do "the bidding of Iran."

US officials are tracking increased efforts by Iranian-backed militias in Iraq, as well as businesses and individuals, to bolster their influence and ability to engage in activities that will get them additional revenue in the wake of oil sanctions being reimposed November 4.

Activities include smuggling drugs, weapons and Iranian oil, which is illegally relabeled as Iraqi and shipped out of the country. The oil shipments so far are small and are not believed to be state-sanctioned.

Iran appears to be trying to create "an armed political wing" inside Iraq with portions both under and not under central Iraqi control, the diplomatic official said.

As for Trump's comments about keeping US troops in Iraq to "keep an eye" on Iran, the official said it was not a mission for the US in Iraq and there was "probably not much" the US could do to keep an eye on Iran from inside Iraq.

CNN's Khalil Abdallah contributed to this story.

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