The massively outnumbered IS fighters are completely boxed in but protected by snipers and thousands of bombs they planted across the city.
That has slowed the progress of the broad alliance of forces battling IS, which is keen to minimise casualties on the way to what would be the biggest victory yet against the jihadists.
Karim al-Nuri, a top leader of the Badr militia and spokesman of the volunteer Popular Mobilisation units, said it would take no more than "72 hours" to liberate Tikrit from IS, which seized it last summer.
Speaking to AFP from the outskirts of Tikrit, near the village of Awja, he said "their number is now 60 to 70".
An lieutenant colonel in the army's elite counter-terrorism forces was more conservative about the battle's evolution, saying "battles in cities are difficult for all armies".
AFP reporters in a northern neighbourhood of Tikrit saw dozens of craters on a single street, caused by the explosion of bombs concealed underneath.
On a roof, a government marksman wearing a white headscarf and who gave his name as Haj Abu Maryam said that by noon, he had already killed two enemy snipers.
Ahmed al-Fraiji said more than 200 rounds had been lobbed into Qadisiya over the past five days alone.
Troops, police, Popular Mobilisation units, Shiite militias and Sunni fighters eager to retake their own city launched the huge assault nearly two weeks ago.
They first cleared outlying areas in Salaheddin province, of which Tikrit is the capital, and broke the city's defences on Wednesday.
Tikrit is the hometown of Saddam Hussein, the remnants of whose Baath party collaborated with IS when it swept across Iraq's Sunni heartland nine months ago.
Military coordination was improved, the cooperation of some Sunni tribesmen secured and Iran is said to have played a key role in the operation's planning and execution.
