The investigation follows a legal complaint brought by a couple who could be the biological parents of the woman's twins.
Four women became pregnant after undergoing the assisted reproduction procedure at the clinic on the same day in December last year.
The couples are only known by the numbers one to four, with couple one having the embryos which are not theirs.
DNA tests are still being carried out to find who the biological parents of the twins are, as couple one must have been given the embryos of one of the three other couples taking part.
The scandal was first reported by the La Stampa daily after a woman who became pregnant found out that the embryos were not biologically hers following a routine genetic test in her third month of pregnancy.
She was quoted by her lawyer Michele Ambrosini saying that she had "doubts" but now wanted to keep them.
"These babies live inside me. I can hear their hearts beating, they are growing and healthy," she said.
The health ministry ordered inspections at the Sandro Pertini hospital in Rome where the procedure took place and a group of experts has been set up to find whether similar mix-ups occurred there in the past.
The woman who filed the complaint later had a miscarriage and is particularly aggrieved, according to her lawyer Pietro Nicotera, although Italian media said even if she and her partner were the biological parents they would have no claim to the children.
Nicotera raised the possibility that the law on assisted reproduction had been violated and said they simply wanted "to remove painful doubts over what might have happened that would last their entire lives".
The clinic has been closed down until further notice.