But he says his government won't be held over a moral barrel when it comes to asylum seeker policy.
Fairfax Media says the women tried to end their lives because they believed their orphaned children would be allowed to settle in Australia.
It spoke to three independent sources who have confirmed the women attempted suicide after being told they would be taken to Nauru or Manus Island.
Mr Abbott said he had not seen the report, but added: "If true, it is a harrowing tale".
However, he said the government would not capitulate to "moral blackmail" when it came to border protection.
"This is not going to be a government which has our policy driven by people who are attempting to hold us over a moral barrel - we won't be driven by that," he told the Nine Network on Wednesday.
"No Australian government should be subjected to the spectacle of people saying `Unless you accept us, I am going to commit self harm'."
The Sydney Morning Herald has quoted Maurice Blackburn Lawyers' Jacob Varghese, who is representing 72 asylum seeker babies, as saying the mothers become distressed when told by immigration officials this week they would never be resettled in Australia because they arrived after July 19 last year.
Arrivals after this date can't be settled in Australia in accordance with legislation enacted by Labor Prime Minister Kevin Rudd.
Mr Varghese has told the paper: "We have heard from our clients there that in the last day several women have attempted suicide or harmed themselves. They are in a state of utter despair."
The Herald says Christmas Island Shire Council president Gordon Thompson has confirmed that there have been women attempting suicide in the detention centre because they believe "if the babies have been born in Australia, they cannot be sent anywhere else".
* Readers seeking support and information about suicide prevention can contact Lifeline on 13 11 14 or Suicide Call Back Service 1300 659 467.
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