Reports say the militants now control at least 60% of the city.
However, air strikes have forced them to pull back from a key government compound where they raised their black flags on Friday.
Ramadi is capital of the country's largest province, Anbar.
The BBC's Ahmed Maher in Baghdad says IS fighters have entrenched their positions around residential districts after beating back Iraqi forces backed up by three regiments sent as reinforcements.
If the entire city were to fall, it would be a major blow to the government, he adds, being just 100km (60 miles) from Baghdad.
IS and Iraqi troops have been battling for months to take control of strategically important Anbar province.
Earlier, the Iraqi army said it had sent three regiments to areas surrounding the city.
Iraqi military spokesman Brig Gen Saad Maan Ibrahim also told Iraqi state television that the US-led coalition had been supporting Iraqi troops with "painful" air strikes since late on Friday.
High-profile Iraqi officials have made several statements to the press and on social media, admitting that the situation is dire, our correspondent says.
It is a sign of how worried the government is, and of how serious the situation on the ground has become, our correspondent added.
The IS assault on the government compound on Friday involved as many as six suicide car bombs. At least 10 police officers were killed, while another 50 were reported to have been taken prisoner.
In response, US Vice-President Joe Biden pledged to deliver heavy weaponry, including shoulder-held rocket launchers and additional ammunition, as well as supplies to the Iraqi forces.
The heavily Sunni province of Anbar covers a vast stretch of the country west from the capital Baghdad to the Syrian border, and contains key roads that link Iraq to both Syria and Jordan.
Iraq's prime minister pledged in April that his forces would "liberate" Anbar from IS after the success of re-taking the central city of Tikrit.
Islamic State (IS) militants have tightened their grip on the centre of the Iraqi city of Ramadi after repelling an advance by government forces.