Singh, a high-profile prisoner on death row for more than 21 years, was attacked by inmates armed with bricks in Lahore's Kot Lakhpat jail on Friday.
He is in intensive care with severe head injuries.
Singh was convicted of spying and carrying out four bomb attacks that killed 14 people in Pakistan in 1990.
His family say he is innocent and merely strayed across the border in Punjab by accident.
Prime Minister Manmohan Singh has described the attack on Singh as "very sad". There have been protests in India over the incident.
Officials from the Indian High Commission based in Lahore visited Singh in hospital late on Sunday, where his condition according to doctors remains "very critical".
A doctor was quoted as saying by the AFP news agency that Singh's family had "been provided a room inside the hospital".
"We want Sarabjit to be taken abroad for treatment," his sister Daljit Kaur told BBC Hindi after visiting the hospital.
"The doctors told us they couldn't say anything about when he would recover. They said, 'Just pray for him'," she said.
India's Foreign Minister Salman Khurshid told the NDTV news channel that the "matter is being taken up at every level" to "ensure that Singh can be sent back to India on humanitarian grounds".
Sarabjit Singh was reportedly attacked as he and other prisoners were brought out of their cells for a one-hour break.
Two inmates have been charged with attempted murder and two officials suspended.
Tensions have increased in the past six months with the execution in India of Kashmiri Afzul Guru over the attack on India's parliament 11 years ago, and of Mohammed Ajmal Qasab, the sole surviving attacker from the 2008 Mumbai attacks.
Sarabjit Singh's lawyer Owais Sheikh told AFP his client had received threats after Guru's execution.
Indian officials and relatives have visited Sarabjit Singh, who is in a coma in Pakistan where he has been held since his conviction for spying.