The government hospital has refuted the allegation.
The affected woman, said to be in her 20s, told a Tamil TV channel on Friday that she underwent the transfusion in April while being pregnant.
She came to know she had HIV only when the hospital informed her after conducting tests in August when she went there for check-up.
The women said though she had been regularly visiting a local public health centre when she was pregnant, she was only administered vaccines there.
"The blood transfusion happened at the Government Kilpauk Medical College and Hospital (GKMCH) only. That is where the mistake happened," she said.
Refuting the allegation, hospital dean Dr P Vasanthamani said the woman was transfused with '100 per cent HIV free' blood.
The woman claimed she had taken up the matter with the state government but to no avail. She had not gone public earlier as her relatives had warned her it will affect her reputation, she said.
She was now coming out because even her relatives were not supporting her anymore, the woman added.
Responding to the woman's allegation, Vasanthamani insisted there was no wrongdoing on the hospital's part and said two units of blood were given on different days in April and both were screened to ensure they were HIV negative.
"As far as we are concerned, we gave only 100 per cent HIV free blood. There is no chance of her contracting the virus here," she told reporters on Friday.
The hospital had record and 'evidence' to show that only HIV negative blood was transfused, she asserted.
After testing positive for the virus, the woman underwent treatment at the hospital for some time, Vasanthamani said.
Her child, who was born recently through a caesarean section, tested negative for the virus, the dean added.
When reporters sought state Health Minister C Vijayabaskar's reaction, he said he came to know about it only from the media and indicated he would respond later.
Earlier, a 24-year-old pregnant woman at Sattur in Virudhunagar district contracted HIV allegedly after being transfused contaminated blood supplied by a blood bank which failed to conduct proper screening for the virus, prompting the Tamil Nadu government to order examination of stocks in the state's blood banks.
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