"Police stormed into Jamia Faridia and arrested dozens of students and shifted them to an unknown place," a senior security official told AFP on Saturday morning.
Since sporadic fighting began on Tuesday, 19 people have been killed.
Dozens of trucks carrying troops moved towards the mosque just before daybreak as gunfire continued amid heavy monsoon rain.
No calls for pre-dawn prayer were heard over the battered mosque's loudspeakers for the second day in a row.
Authorities have forced journalists to move away from the area, an AFP photographer said.
Early on Saturday fresh gunfire and blasts erupted at the Red Mosque, or Lal Masjid, after a seven-hour lull, but officials said security forces had not stormed the fortified complex of buildings.
"We are not making any advances, this is part of the operation to secure the release of women and children held hostage by the cleric," a security official said.
Students from the mosque threw grenades and petrol bombs at security forces, who responded by firing, he said.
The government had earlier rejected a conditional surrender offer by Ghazi, whose brother Abdul Aziz, the founder of Faridia seminary, was captured on Wednesday while trying to flee the mosque dressed in a burqa.
"We have decided that we can be martyred but we will not surrender. We are ready for our heads to be cut off but we will not bow to them," Ghazi told the private Geo television station.
The cleric later gave a sermon to his students at traditional Friday prayers in which he said they had already written their wills.
One Pakistani official said four students were killed in the clashes on Friday when a mortar fired by security forces hit a room in the mosque.
Earlier on Friday, extremists shot and wounded a man wanting to see his daughter inside an Islamic school attached to the mosque, officials said.
Hundreds of Islamic students are still inside the mosque compound, along with up to 60 "hardcore" armed students, officials have said.