Al-Qaeda-linked militants now control the south of the city. An Iraqi reporter there says tribesmen allied with al-Qaeda hold the rest of Fallujah.
Fighting there erupted after troops broke up a protest camp by Sunni Arabs in the city of Ramadi on Monday.
They have been accusing the Shia-led government of marginalizing the Sunnis.
The recent fighting in Fallujah and Ramadi has pitted government troops on the one hand and the Islamic State of Iraq and the Levant (Isis), which is affiliated with al-Qaeda, and Sunni tribesmen on the other.
Local Sunni Arabs have been angered by they perceive as discrimination by the government of PM Nouri Maliki.
They also say their minority community is being targeted by anti-terrorism measures implemented to stem the surge in sectarian violence.
Late on Saturday, eyewitnesses said al-Qaeda militants were in evidence on the streets of Fallujah, riding pick-up trucks mounted with anti-aircraft guns.
Reuters news agency said saying troops were shelling parts of the city to try to retake it.
Iraqiya state television quoted Nouri Maliki as saying: “We will not back down until we end all terrorist groups and save our people in Anbar.”
On Tuesday, he agreed to withdraw the army from towns and cities in Anbar province, to allow police to resume control of security.
But as soon as soldiers left their posts, militants aligned to al-Qaeda appeared in Ramadi, Fallujah and Tarmiya, storming police stations, freeing prisoners and seizing weapons.
Nouri Maliki reversed his decision the next day, sending soldiers back to Anbar.
But on Thursday, militants in Ramadi and Fallujah raised black flags on buildings and used the loudspeakers of mosques to call on people to join their struggle and support a “peaceful takeover”.
In recent months Sunni militants have stepped up attacks across Iraq, while Shia groups began deadly reprisals – raising fears of a return to full-scale sectarian conflict.
On Wednesday, the UN said at least 7,818 civilians and 1,050 members of the security forces had been killed in 2013.
RM7OW-7hLikImage source: Wikimedia Commons President-elect Donald Trump celebrated his election victory at the Ultimate Fighting…
Millions of voters across the US chose to return Donald Trump to the White House…
Donald Trump declares victory in the US election as he addresses jubilant supporters in Florida.…
Stocks around the world are rising as Donald Trump appears to be on the cusp…
Donald Trump has won Pennsylvania, North Carolina and Georgia and taken a lead over Kamala…
Quincy Jones, the celebrated musician and producer who worked with Michael Jackson, Frank Sinatra, Ray…