Early Show national correspondent Hattie Kauffman says the mother is in her mid-thirties and lives with her parents.
There's been no mention of the octuplets' father, Kauffman observes.
The grandfather, she adds, is apparently going to head back to his native Iraq to earn money for the growing family. He told CBS News he's a former Iraqi military man.
Kauffman reported Thursday, and the octuplets' maternal grandmother now confirms to the Los Angeles Times, that.
And a family acquaintance had told Kauffman that two of the six other kids are twins, and the six range in age from about two to about seven.
The mother's name is still being kept under wraps.
But her mother, Angela Suleman, also tells the newspaper her daughter conceived the octuplets through a fertility program.
Suleman told the Times her daughter had embryos implanted and, "They all happened to take."
On The Early Show Friday, the scientific director of an Atlanta-area fertility clinic blasted whichever clinic did the implantations, saying he's "stunned."
Doctors at the hospital where the octuplets were born, Kaiser Permanente Bellflower Medical Center in Bellflower, Calif., some 17 miles southeast of L.A., say the patient came to them already three months pregnant.
Asked at a news conference whether fertility assistance should be provided for a mother who already has multiple children, Dr. Harold Henry, part of the team that delivered the octuplets, said, "Kaiser has no policy on that, adding that doctors counseled the woman on her options.
"The options," said Henry, "were to continue the pregnancy or to selectively abort. The patient chose to continue the pregnancy."
Dr. Karen Maples, who also helped deliver the octuplets, read a statement from the mother saying, "My family and I are ecstatic about all of their arrivals."
The woman and her children live in a neighborhood of small, one-story homes, Kauffman reports, all with two-to-three bedrooms at most. Soon, she pointed out, there will be 14 children and at least three adults living in one of the homes -- until the grandfather heads back to his native Iraq,
Kauffman says unanswered questions include where the woman got the fertility treatments and how they were paid for.
On The Early Show Friday, Michael Tucker, scientific director of Georgia Reproductive Specialists, says all these developments leave him "stunned. As the story's unfolded and it's gone from the potential use of just fertility drugs, or misuse thereof, to actual, apparently, IVF (in-vitro fertilization) with transfer of embryos, this is just remarkable to me that any practitioner in our field of reproductive medicine would undertake such a practice."
Tucker, who has a doctorate in reproductive physiology, says it's "absolutely" possible the octuplets' mother got pregnant with them by taking fertility drugs on her own without the help of a clinic, "and that seemed the most plausible scenario, simply because the profession, we're policed by the American Society of Reproductive Medicine, has focused so minutely on the fact that we need to reduce the number of embryos that we transfer. We really are all about seeking the one, the one embryo that's going to make the healthy, single-born baby.
"And this kind of multiple plethora excess of babies is too much of a good thing. And it's rather a slap in the face of the whole profession, simply because it's going in the wrong direction.
"And it's unfortunate, because the media pick up on this and seem to go, I think, Arthur Kaplan from UPenn (University of Pennsylvania) said the media tend to go goo-goo gaga over this and, in fact, it's really a bit of a medical disaster."
"Had she walked into a fertility clinic and said, 'Listen, I've got other children, the oldest seven, the youngest two,' co-anchor Julie Chen asked Tucker, "is there any ethical responsibility on the clinic's part to say, 'I'm not going to treat you,' or, 'You know what? This is not a good idea?" '
"Suffice to say," Tucker responded, "I've been in this business for 25 years now. And it's pretty much standard practice in all clinics to have some form of psychological evaluation of the patient. Also, their sociological circumstances. And I'm stunned, actually, that a clinic would proceed to treat a patient in this circumstance and then even to get to perhaps the transfer of embryos and ponder the transfer in, I believe, the lady's mid-30s, a 35-year-old -- she should be receiving two embryos, maximum, as a transfer into her uterus to have had eight transferred is somewhat -- is extremely irresponsible."