As I recall: For the first six volumes of that series, it was incredibly unclear what the correct answer to your question should be. As you said, we gradually learned that other people at the White Tower sometimes made remarks which suggested there was a widespread perception that members of the Green Ajah in particular often were romantically involved with their own Warders. And I believe Lan stated to Nynaeve that Aes Sedai rarely married, a choice of words which suggested it had been known to happen, at least a few times, during the past three thousand years. So the strong implication was that an Aes Sedai, whether Green Ajah or otherwise, was capable of having a love affair (or even a marital relationship) with a man without getting kicked out of the White Tower for breaking an ironclad law. In other words, it was not mandatory for Aes Sedai to live in the strictly celibate way that real-world nuns are expected to live. (Although I wouldn't be surprised if many of them did so, by choice.)
But was the typical Aes Sedai physically capable of getting pregnant as a result of intimate interludes with a man? Whether her own Warder, or someone else entirely? If I recall correctly: In the first six books, Robert Jordan never wrote anything which stated or heavily hinted that the answer to that question was: "Yes, it can happen if an Aes Sedai wants it to -- but she usually doesn't!" On the other hand, in those first six, he never wrote anything which committed him to the idea that the answer was: "No, the Aes Sedai have known for many centuries that it is impossible for one of them to get pregnant, no matter how hard she tries."
In other words, Jordan was being extremely careful not to address that subject until he was good and ready. It was clear that some (certainly not all!) of the Aes Sedai slept with their Warders (or possibly with other men) from time to time, and it was notable that the scenes set at Tar Valon contained a total absence of any mention of some of those Aes Sedai being "working mothers" with their offspring sharing their living quarters in the White Tower (or being put up for adoption, or whatever), but the vast lack of kids could have been due to any one of the following:
An inherent sterility which was a side effect of being an Aes Sedai.
A "spell" (i.e. a "weave," as Jordan would put it) which any trained channeler of saidar could easily use on herself, over and over, as a birth control method which was guaranteed 100 percent accurate, no matter what she was doing for fun in her spare time.
A more mundane method of birth control which was extremely reliable and didn't involve the One Power at all.
As far as we knew at first, any of those three theories might explain why no Aes Sedai was being described to us as having ever been pregnant with a child.
Note: Eventually -- in Book 9, Winter's Heart, Chapter 12 -- it was established that in the region of the world where the main action takes place, and probably all over the rest of the planet, there was an herbal preparation which was considered extremely effective in helping a woman avoid pregnancy if she didn't want one in the near future. Something called "heartleaf tea," which was evidently the local equivalent of "the Pill." The implication was that all women were aware of heartleaf's properties. But that factoid, by itself, would not answer your question about whether or not Aes Sedai in particular can become pregnant if they want to.
Fortunately, that particular question had already been answered two books earlier.
In the seventh volume -- A Crown of Swords, Chapter 36 -- two skilled Healers both look at a nasty wound in Rand's side. Each wants to help him recover from it, if they can. One is a channeler of saidar: Samitsu Tamagowa, whom Cadsuane believes to be the finest Healer in the entire Yellow Ajah. The other is a recently-trained channeler of saidin: Damer Flinn of the Asha'man, who seems to have a special instinct for healing weaves. (Which is lucky, because most Asha'man don't, and neither does Rand, and so there's no one qualified to teach Damer very much in this field. He has to work on hunches and experimentation, apparently.)
Damer examines Rand's wound, and describes what he perceives in there in such terms that Samitsu (although not the viewpoint character in this scene) is obviously convinced that he does, in fact, know what he's talking about. Then Damer does something to the wound that Samitsu has no clue how to do. Since Damer is channeling saidin, Samitsu cannot see just what he does, even though she's watching the whole thing, but her own Healing abilities do allow her to detect a significant difference between the "before" and the "after" where Rand's medical condition is concerned. After she's sure Damer really accomplished something unprecedented in her experience, here is her reaction:
To Flinn's evident surprise, Samitsu rounded the bed to help him
rise. "You will tell me what you did," she said, regal tone at strong
odds with the way her quick fingers straightened the old man's collar
and smoothed his lapels. "If only there was some way you could show
me! But you will describe it. You must! I will give you all the gold I
possess, bear your child, whatever you wish, but you will tell me
all that you can."
That part about "bear your child" is the key part for our purposes.
Samitsu was never revealed to be Black Ajah, and thus, as with most of the woman calling themselves "Aes Sedai" at that time in the series, we can presume that she had been rigorously bound by the Three Oaths ever since she was first raised to the shawl at Tar Valon.
Therefore, Samitsu was incapable of deliberately making any false statement about anything. That means she had to be sincere when she made that offer -- not getting carried away with outrageous hyperbole. Bearing a man's child had to be something that she knew, beyond a reasonable doubt, she was physically capable of doing -- if she felt properly motivated to take the trouble!
In other words: Samitsu must have been telling the literal truth when she suggested that if Damer Flinn could teach her something very useful about the fine points of healing wounds with the One Power, something that no woman knew how to do, then Samitsu would be willing to share a bed with Damer until she was sure he had gotten her pregnant, and then she'd spend the next nine months carrying the baby to term. Assuming that Damer could first hold up his end of the bargain by teaching her something really useful, and assuming he insisted that bearing his child was the price she had to pay for special tutoring which no one else could provide.
Note: To do him justice, Damer evidently never even considered trying to take her up on that offer. He was clearly surprised by that speech I quoted, but he was perfectly willing to compare notes with Samitsu on Healing issues without needing to be bribed with sexual favors or anything else.