BAGHDAD, July 7 (Reuters) - U.S. diplomats and troops in Iraq and Syria were targeted in three rocket and drone attacks in the past 24 hours, U.S. and Iraq officials said on Wednesday, including at least 14 rockets hitting an Iraqi air base hosting U.S. forces, wounding two American service members.

While there were no immediate claims of responsibility for the attacks - part of a wave targeting U.S. troops or areas where they are based in Iraq and Syria - analysts believed they were part of a campaign by Iranian-backed militias.

Iraqi militia groups aligned with Iran vowed to retaliate after U.S. strikes on the Iraqi-Syrian border killed four of their members last month.

Two people were slightly wounded in the rocket attack on the Ain al-Asad air base in western Iraq, said coalition spokesman U.S. Army Colonel Wayne Marotto. The rockets landed on the base and its perimeter. He said earlier that three people were wounded.

U.S. officials, speaking on condition of anonymity, said the two personnel injured were U.S. service members. One suffered a concussion and the other had minor cuts, one of the officials added.

Two rockets were fired at the U.S. Embassy inside Baghdad's Green Zone early on Thursday, Iraqi security sources told Reuters.

The embassy's anti-rocket system diverted one of the rockets, said one of the sources - a security official whose office is inside the Green Zone. The second rocket fell near the zone's perimeter, security officials said.

Sirens blared from the embassy compound inside the zone, which houses government buildings and foreign missions, the sources said.

In Syria, the U.S.-backed Syrian Democratic Forces said no damage was done by a drone attack on the Al Omar oil field in an eastern area bordering Iraq where U.S. forces came under rocket fire but escaped injury on June 28.

The Pentagon said a drone had been brought down in eastern Syria and that no U.S. service members had been injured and there had not been any damage.

Iraqi army officials said the pace of recent attacks against bases hosting U.S. forces with rockets and explosive-laden drones was unprecedented.

Iraqi military sources said a rocket launcher fixed on the back of a truck was used in Wednesday's attack and was found on nearby farmland set on fire.

On Tuesday, a drone attacked Erbil airport in northern Iraq, targeting a U.S. base on the airport grounds, Kurdish security sources said.

Three rockets also landed on Ain al-Asad on Monday without causing casualties.

ESCALATION

The United States has been holding indirect talks with Iran aimed at bringing both nations back into compliance with the 2015 Iran nuclear deal, which was abandoned by then-President Donald Trump. No date has been set for a next round of the talks, which adjourned on June 20.

Hamdi Malik, an associate fellow at the Washington Institute and a specialist on Iraq’s Shi’ite militias, said the attacks were part of a coordinated escalation by Iranian-backed militias in Iraq.

The attempt to strike in eastern Syria appeared to be the first example of operations being carried out simultaneously in both countries.

"It seems to me they have the green light from Iran to escalate, especially given that the nuclear negotiations are not going well. But at the same time, they do not want to escalate beyond a certain point – they are more vulnerable to U.S. air strikes than they used to be - and they don’t want to overcomplicate the negotiations Iran is holding with the West."

The United States told the U.N. Security Council last week that it targeted Iranian-backed militia in Syria and Iraq with air strikes to deter them and Tehran from conducting or supporting further attacks on U.S. personnel or facilities.

Iran denied supporting attacks on U.S. forces in Iraq and Syria and condemned U.S. air strikes on Iranian-backed groups.