IS captured the city in May, in what was a major embarrassment for Iraq’s weak central government.
Iraqi forces began an offensive to retake the Sunni Muslim city in early November and say they have now reached the centre, where an estimated 300 militants are stationed.
The military operation has taken some time because the Iraqi government has opted to use its own troops, without help from Shia militias backed by Iran. The US had advised Iraq that relying on Shia troops to fight hardline Sunni militants would inflame sectarian tensions.
Iraqi military spokesman Sabah al-Numani said: “Our forces are advancing toward the government complex in the centre of Ramadi. The fighting is in the neighbourhoods around the complex, with support from the air force.”
Capturing Ramadi, 60 miles west of the capital Baghdad, was a coup for IS extremists.
Iraqi forces crossed the Euphrates river early this morning, using a bridge that had been destroyed by IS, but repaired by army engineers.
Mr al-Numani said: “Crossing the river was the main difficulty. We’re facing sniper fire and suicide bombers who are trying to slow our advance. We’re dealing with them with air force support.”
If Ramadi is retaken, it would be the second major city after Tikrit to be captured from IS in Iraq and would provide a psychological boost to Iraqi forces. Iraqi Kurdish forces retook Sinjar from IS last month.
IS controls Mosul, Iraq’s second largest city, and Falluja, between Ramadi and Baghdad.
The US, Britain and several other countries are bombing IS positions in Iraq and neighbouring Syria.