Death row prisoner Amer Aftab tried to attack Sarabjit a few days ago and abused him, the sources told PTI.
The reason for the altercation could not immediately be ascertained.
Aftab, who was being held in a barrack near Sarabjit's barrack in one of the most secure sections of Kot Lakhpat Jail, was among six prisoners who attacked the Indian national at about 4.30 pm.
The prisoners gathered near Sarabjit's barrack after evading guards, a jail official said on condition of anonymity.
The prisoners overpowered two jail wardens and Aftab snatched the key of Sarabjit's cell and opened the door.
Sarabjit was hit on the head with bricks and his face, neck and stomach were cut with blades and sharp pieces of a ghee tin, at least two sources told PTI.
"When the staff found Singh, he was unconscious and bleeding profusely. The staff removed his clothes which were covered with blood. They put on the shirt and pants of a jail warden and took him to Jinnah Hospital," the jail official said.
Earlier, unnamed officials of Kot Lakhpat Jail were quoted by Pakistani TV news channels as saying that only two prisoners Amer Aftab and Mudassar were involved in the attack.
The officials were also quoted as saying that Sarabjit was attacked when he and other prisoners were brought out of their cells for an hour-long break.
However, the sources rejected these claims and said Sarabjit was assaulted by at least six prisoners within his barrack.
Aftab, who was sentenced to death for murdering a mayor's brother, and Mudassar were locked up after the attack and interrogated, sources said.
Malik Mubashir, Deputy Superintendent of Prisons in Punjab province, was appointed to head the inquiry into the attack.
Mubashir would probe all aspects of the incident and ascertain whether some jail officials may have been involved or linked to the attack on Sarabjit.
He will also ascertain whether the attackers were motivated by "religious elements", the sources said.
Sarabjit was admitted to the state-run Jinnah Hospital with a severe head injury this evening and his condition was described as critical.
Authorities had tightened Sarabjit's security after the recent execution in India of Afzal Guru, who was convicted for his involvement in the 2001 terror attack on the Indian parliament.
In January, another Indian prisoner in Kot Lakhpat Jail, Chambail Singh, died after allegedly being assaulted by prison staff.
Though he died in mysterious circumstances, Singh's autopsy was performed almost two months after his death.
Sarabjit was convicted for alleged involvement in a string of bomb attacks in Punjab province that killed 14 people in 1990.
His mercy petitions were rejected by the courts and former President Pervez Musharraf.
The outgoing Pakistan People's Party-led government put off Sarabjit's execution for an indefinite period in 2008.
Sarabjit's family says he is the victim of mistaken identity and had inadvertently strayed across the border in an inebriated state.