Parts of Ramadi, the provincial capital, and all of Fallujah to its east, fell to anti-government forces in January.
In June, IS-led militants began overrunning more of Anbar and have gained further ground in recent weeks, raising fears that the province, which stretches from the borders with Syria, Jordan and Saudi Arabia to the western approach to Baghdad, could fall completely.
"IS launched a surprise attack from four directions -- north, west, east and south of Ramadi," a police first lieutenant in the city told AFP by telephone today.
Police Captain Qusay al-Dulaimi said "the mortar fire has been continuous since midnight."
Unlike in the past, mosque loudspeakers called on people to fight IS rather than resist government forces.
Soldiers, police and tribal fighters were able to retake one area the militants had seized and hold off attacks on others, deputy provincial council chief Faleh al-Essawi and tribal leader Sheikh Rafa Abdulkarim said.
But the sound of gunfire could still be heard in the city early this evening, indicating that fighting in the area was not over.
Essawi said 12 militants died in the fighting.
A fresh spate of attacks in recent weeks has seen the jihadists extend their grip over the province, where only a handful of pockets remain under the control of Iraqi security forces backed by Shiite militias and Sunni tribal fighters.
But there is still little threat of a direct assault on Baghdad, as militants would have to gain much more ground for one to be viable.
Meanwhile, in the northern town of Zab, IS gunmen publicly executed two young men Friday for allegedly cooperating with security forces.
Iraqi security forces performed poorly when faced with the initial jihadist-led onslaught in June, with several divisions collapsing in the north.
