Refugee advocate Pamela Curr, who had met the man on several occasions, told SBS News he had been in MITA for "around two years" and had likely become "totally hopeless".
Ms Curr said the 23-year-old was a "gentle, lovely Hazara man" who had fled "political violence" in Afghanistan.
She said he arrived in Australia by boat and was on a bridging visa. When this expired, he became "destitute .. and was sleeping in his car" before being detained.
Advocates say he was taken to hospital after the incident, with a spokesperson for Ambulance Victoria confirming to SBS News paramedics were called to Broadmeadows at about 4.30am on Monday.
The Refugee Action Coalition claimed the 23-year-old was now under guard in the high security section in MITA.
"Just a reminder that the government has claimed that immigration detention is administrative and not punitive in nature," the group tweeted.
The latest act of self-harm has prompted warnings of another death in detention.
"I refuse to believe that the only way to stop 'deaths at sea' is to have deaths on land. Australia’s refugee policy is monstrous," Greens MP Adam Bandt tweeted after reports of the self-harm emerged.
SBS News has sought comment about the reports from the Department of Home Affairs.
Readers seeking support and information about suicide can contact Lifeline 24 hours a day online and on 13 11 14. Other services include the Suicide Call Back Service on 1300 659 467, Beyond Blue and Kids Helpline (for people aged five to 25) on 1800 55 1800.
More information about mental health is available at Beyond Blue.