ABOARD A U.S. MILITARY AIRCRAFT » U.S. defense officials said Thursday that Russia has pulled most of its forces away from the Ukraine border, a withdrawal that the U.S. has been demanding for weeks.

They said about seven battalions remained, amounting to a couple of thousand troops. U.S. officials had estimated as many as 40,000 Russian forces had been aligned along the border with a restive eastern Ukraine that has been wracked with violence between government security forces and pro-Russian separatists.

The defense officials spoke only on condition of anonymity because they were not authorized to speak publicly about the precise numbers.

Defense Secretary Chuck Hagel didn’t provide any details to reporters traveling with him at the start of a 12-day overseas trip, but he called the withdrawal promising.

"They are not where they need to be and won’t be until all of their troops that they positioned along that border a couple of months ago are gone," Hagel said.

"We do know that thousands of Russian troops have been pulled back and are moving away. But we also know that there are still thousands of Russian troops still there that have not yet moved," Hagel said.

Hagel said he has not spoken to his Russian counterpart about the withdrawal.

Hagel was among Obama administration officials who expressed new concerns Thursday about rising violence in eastern Ukraine, including the downing of a military helicopter by pro-Russian rebels.

The White House and State Department both said a de-escalation of the crisis was imperative and called on Russia to exert pressure on the separatists to get them to end the fighting and release a group of international monitors who have been detained in eastern Ukraine since earlier this week.

"We are disturbed by the ongoing violence in eastern Ukraine," presidential spokesman Jay Carney said at the White House. While the U.S. has not been able to verify what happened to the helicopter, he said, "We are concerned that this indicates separatists continue to have access to advanced weaponry and other assistance from the outside."

Ukraine’s acting president said earlier Thursday that 12 troops died when rebels shot down a military helicopter in Slovyansk using a portable air defense missile.

Even before the incident, Secretary of State John Kerry spoke to his Russian counterpart, Sergey Lavrov, on Wednesday to reiterate U.S. concerns about the deteriorating situation in Ukraine, the State Department said.

Kerry raised with Lavrov reports of Chechen fighters crossing into Ukraine to join the separatists, spokeswoman Jen Psaki said at the State Department.

Kerry "pressed Foreign Minister Lavrov to end all support for separatists, denounce their actions and call on them to lay down their arms," she said.

"Our broad view, as you know, is that de-escalation is the proper path forward," Psaki added, although she said she was not aware of concerns that Ukrainian security forces were using disproportionate means to quell the fighting as some Russians have alleged.

Carney and Psaki also said it was unacceptable that insurgents have detained four observers from the Organization for Security and Cooperation in Europe. They demanded their immediate release.

The U.S. has called on Russia repeatedly to help de-escalate tensions in Ukraine, including withdrawing troops massed near Ukraine.

Associated Press writers Matthew Lee and Josh Lederman in Washington contributed to this report.