(Adds Pentagon comment)

BAGHDAD, June 21 (Reuters) - Iraq's armed forces pressed on with their offensive to retake the city of Falluja from Islamic State on Tuesday, dislodging the militants from two eastern districts and pushing them back into a handful of northern and western neighbourhoods.

A military statement said the elite counter-terrorism forces took the northeastern Shurta police district while units from Baghdad operations command recaptured the Askari military neighbourhood.

Islamic State still held the northern districts of Jughaifi and Golan as well as the western banks of the Tigris river.

Fighting to recapture the Iraqi city longest held by Islamic State, now in its fifth week, has forced more than 85,000 residents to flee to overwhelmed government-run camps.

Iraq's Prime Minister Haider al-Abadi said last week that the recapture of Falluja would pave the way for the military to march on Mosul, Islamic State's de facto capital.

The operation's commander Lieutenant General Abdul Wahab al-Saidi told Reuters on Monday that U.S.-backed Iraqi forces expected the final push would take place in the west.

The Iraqi army is also pressing on with an offensive to dislodge the militants from farmlands located on the western bank of the Euphrates, opposite the built-up area of Falluja, that the militants use as a base for snipers and mortar attacks.

"The simultaneous attacks are continuing from four directions to tighten the noose around Daesh (Islamic State)fighters entrenched in houses among civilians and prevent them from catching their breath," said Colonel Ahmed al-Saidi from the federal police.

A Pentagon spokesman in Washington said that about a third of the city had been cleared of Islamic State forces and the U.S.-led coalition had carried out 85 air strikes on the city in the past four weeks.

"Iraqi forces continue to advance at a rate of about five to 10 percent of the city per day being careful to safeguard the lives of civilians and limit property damage to the extent possible," Peter Cook said.

Government troops launched the operation on May 23 to retake Falluja, a bastion of the Sunni Muslim insurgency against U.S. forces that toppled Saddam Hussein, a Sunni, in 2003, and later against Shi'ite-led governments.

The militants seized Falluja in January 2014, six months before they declared a "caliphate" over part of Syria and Iraq. (Reporting by Ahmed Rasheed. Additional reporting by Idrees Ali; Writing by Maher Chmaytelli; Editing by Richard Balmforth and Diane Craft)

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