Esper's denial came after multiple U.S. media outlets reported on Monday that the USA would be repositioning forces within weeks, suggesting movement out of Iraq.
Two Iraqi bases housing United States and coalition troops were targeted, one at Al Asad and one in Irbil, at about 02:00 local time on Wednesday (22:30 GMT on Tuesday).
"In order to conduct this task, Coalition Forces are required to take certain measures to ensure that the movement out of Iraq is conducted in a safe and efficient manner", said the letter.
Gen Milley acknowledged that some language in the letter "implies withdrawal", but that "is not what is happening". Abdul-Mahdi says the only problem they had was that the English and Arabic versions were not identical and they want clarification of which was right.
Mr Esper said the USA remains committed to the campaign to defeat the so-called Islamic State group in Iraq and the wider region.
The UK has called for an urgent deescalation as tensions continued to mount in the Middle East after the top Iranian general was assassinated by a United States drone on Friday.
The attack came in response to a U.S. terrorist drone strike days earlier that martyred one of Iran's top Generals, the former Commander of IRGC's elite Quds Force, Lieutenant General Qassem Soleimani.
Abdel-Mahdi, who is also Iraq's general army commander, has been holding talks aimed at de-escalation and avoiding an "open war" that would "primarily" harm Iraq and the region, the spokesman said in a statement, carried by Iraq's state news agency INA.
LSU linebacker Michael Divinity Jr reinstated for title game
It's been a season-long run of performances even coach Ed Orgeron has said it's hard to step back and truly appreciate. Burrow's current pace projects him to finish with 5,580, which would be good for No. 5 all time in a single season.
Tom Brady Responds To Retirement Talk After New England Patriots' Playoff Defeat
League-leading rusher Derrick Henry and his Titans team will travel to Foxborough in the late game on Wild Card Saturday. Brady ended what may be his infamous final scene completing 20 of 37 passes for 209 yards and no touchdowns.
Prince Harry and Meghan Markle Owe Their Canadian Getaway to David Foster
Royal fans were left delighted earlier this week when Meghan Markle and Prince Harry shared a new photo of baby son Archie . The photo of Prince Harry cuddling his son was posted on the royal couple's Instagram account to mark the New Year.
But Pentagon Joint Chiefs Chairman Mark Milley said the letter was only a "draft" that shouldn't have been sent.
"We are not leaving Iraq and a draft unsigned letter does not constitute a policy change", Esper told reporters at the Pentagon.
The letter appeared to have been sent by Brig Gen William H Seely, head of the U.S. military's task force in Iraq, to Abdul Amir, the deputy director of Combined Joint Operations.
On Sunday, Iraq's parliament passed a resolution telling the government to end the presence of all foreign troops. "The Secretary General noted that the suspension of North Atlantic Treaty Organisation training activities is a temporary force protection measure and that North Atlantic Treaty Organisation is prepared to continue its activities when the situation permits", the press release from PMO said.
"So in the future, what should we do if we get such a request?"
Earlier Monday, Abdul-Mahdi met with US Ambassador Matthew Tueller, telling him that the two countries need "to work together to execute the withdrawal of foreign troops from Iraq".
An initial Arabic translation of the letter had contradicted the English version, he added, and another copy was then resent.