There isn't really any evidence, but there is one myth about the possibility of children. After Henry VIII died in January 1547, Elizabeth was sent to live with his surviving wife, Katherine Parr. Katherine had re-married to Edward VI's uncle, Thomas Seymour, who was a notorious womanizer. Seymour didn't go as far to court the 13/14 year old Elizabeth, but there are records where he had behaved in not so appropriate a manner. He would go into her bedroom before she had got got dressed in the morning, and would tickle her and chase her around the room. Katherine occasionally joined him, and her governess, Kat Ashley, would half-heartly protest, but she mostly found it amusing. It was reported that Katherine found Seymour and Elizabeth kissing while she was 6 months pregant, and sent Elizabeth away to live with Sir Anthony Denny. Katherine died in childbirth, and the baby Mary has no other historical mention, but while she was dying from her fever, she was uttering that Seymour had poisoned her in order for him to marry Elizabeth. This formed the basis of his arrest, and once he broke into Edward's chambers, he shot the kings dog, and was then arrested, tried for treason, denied all reports of him and Elizabeth, and was executed in 1549. Elizabeth had made no public appearances after she moved to the Denny's, and was interrogated for her role in the suspected marriage proposal. It is this suspected affair, and her reclusiveness from court activities for 18 months, that has fueled it. It could very well be possible that Seymour and Elizabeth had sex, and she became pregnant, and that was why she was sent away and explains her absence. However, there is no sufficient evidence to say this, or even if the child survived. Alison Weir, historian and historical novelist seems to think there is some substance to this myth. In her novel 'The Lady Elizabeth', then Elizabeth is pregnant by Seymour. In truth, there is only evidence of a suspected affair, and nothing else.
Along with this, there has been a rumour that she fell pregnant by her favourite, Robert Dudley, Earl of Leicester. Their love and courtship of each other is well known, they had known eachother since childhood, and were imprisoned in the Tower by Mary I at the same time. We dont know if they slept together at all, even at the time it was nothing but rumours, and if they did, then they were very careful. It is highly unlikely that Elizabeth had a child by Dudley. First they have had to have had sex without anyone finding out, second, it would have been extremly difficult and practically impossible for Elizabeth, a naturally silm woman, to hide a growing baby bump from the court, let alone all the ladies that attended to her every second of the day. Third, it would have impossible for her to have gone through labour with the entire court knowing about it.
The Seymour myth has some small foundations, but even that is highly doubtful