WASHINGTON (Reuters) - Iranian-backed militia pose a “significant” threat to U.S. forces in Iraq, the State Department’s top diplomat for the Middle East said on Thursday, about a week after U.S. President Donald Trump warned of an attack by Iran or its proxies.

David Schenker, assistant secretary of Near Eastern Affairs, did not give details about the threat, but told reporters in a teleconference: “It continues to be significant.”

Iran-backed paramilitary groups have regularly been shelling bases in Iraq that host U.S. forces, and the area around the U.S. embassy in Baghdad.

On Monday, three Katyusha rockets landed near a district in southern Iraq that houses workers for foreign oil companies, including U.S. oil service company Halliburton. No casualties or damage were reported.

Last week, Trump said Iran or its proxies planned a sneak attack on U.S. targets in Iraq, and warned they would pay a “very heavy price” but gave no details.

U.S.-Iranian relations have been bitter since the Islamic Revolution toppled the U.S.-backed shah of Iran in 1979 and ushered in an era of theocratic rule.

While there was a detente with the 2015 Iran nuclear deal, relations have deteriorated with Trump’s decision nearly two years ago to abandon that international agreement and reimpose U.S. sanctions that have crippled the Iranian economy.

Worsening tensions, a Jan. 3 U.S. drone strike in Iraq killed Qassem Soleimani, the head of Iran’s elite Quds Force. It also killed Abu Mahdi al-Muhandis, who founded Iraq’s Shi’ite Kataib Hezbollah militia after the 2003 U.S.-led invasion.

Earlier this week, U.S. Secretary of State Mike Pompeo said the United States proposed a ‘Strategic Dialogue’ with Iraq to be held in June, a bid to restore strained bilateral ties.

Schenker, who in March said Washington was “enormously disappointed” by Iraq’s performance in protecting U.S.-led coalition forces, repeated Washington’s concerns.

“It is on the Iraqis - if they value that relationship - to take certain steps and that includes providing protection to the coalition forces who are in Iraq, if they want those forces to remain,” Schenker said.

On Thursday, Iraq’s president named intelligence chief Mustafa al-Kadhimi as prime minister-designate, the third person tapped to lead Iraq in just 10 weeks as it struggles to replace a government that fell last year after months of deadly protests.

Schenker said of Kadhimi: “If Kadhimi is an Iraqi nationalist, dedicated to pursuing a sovereign Iraq, if he is committed to fighting corruption, this would be great for Iraq, and we think it would be great for our bilateral relationship.”