To begin answering this question, it's important to think about the common law practice used in British colonies and the US, partus sequitur ventrem. This Latin phrase established the practice of enslaved people's status following the lineage of the mother - instead of the father. This meant in practice that any child born of an enslaved woman was born into slavery.

Another key thing to remember when handling any question on slavery in the US, it's important to remember that there are few primary sources from enslaved people's points of view that exist. WPA interviews held during the FDR era have received some fair criticisms, however they do offer something when coupled with the slave narratives that were published at the time of slavery.

Now, dealing specifically with the situation you've enquired, it would help to break down the surrounding issues here. Let's begin:

First off, we know that enslaved women gave birth to the children of their white masters through the response that the Plantation mistresses had to these events. We see multiple examples of this, one that sticks out is in Out of the House of Bondage by Thavolia Glymph. She writes on Gertrude Thomas' experience as a mistress on a plantation writing,

How else could she account for the 'white children of slavery' who lived in her midst? Her maid Lurany's child was 'as white as any white child.' Plantation mistresses recognized these children and the strains it had on their marriage. Glymph later refers to Jacob Manson, who; Remembered a mistress who was reported to have shot her husband dead in self-defense following a quarrel about his relationship with a female slave..."She grabbed de gun an let him have it. She shot 'im dead in de hall. Dey had three chillun, two songs and one married daughter...De married daughter an her husband took charge of de place. Missus an her sons never come back as I knows of."

Second, I'd like to reshape how we define, "the child of a slaveowner." In Rethinking Rufus (not a plug, but this book just came out May 1st of this year and I highly recommend it) Thomas A. Foster writes on sexual violence inflicted upon enslaved men. One of these violent acts is the act of "coupling" where slave masters would force enslaved men and women to essentially breed to produce more slaves. Foster writes that;

Men who were forced to reproduce bore the responsibility of ensuring women's compliance and had to negotiate this dynamic...In all cases, men's actions were supported in the background by the threat of punishment against the women if they resisted - and in the back of men's minds was their own punishment for disobeying a master's directive to reproduce.

This should be considered in this question because it speaks to the overarching issue of enslaved women giving birth to their slaveowner's child: Enslaved women (and men) were seen as property, and that their reproductive abilities were in and of themselves ways to make more money.

Deborah Gray White writes about the sexual exploitation of enslaved women when she writes in Ar'n't I a Woman?;

Once slaveholders realized that the reproductive function of the female slave could yield a profit, the manipulation of procreative sexual relations became an integral part of the sexual exploitations of female slaves...This was particularly true after Congress outlawed the overseas slave trade in 1807.

Ultimately, childbirth among enslaved women was actively encouraged for the simple reason of maintaining enslaved people's numbers, and maximizing profits. Whether that profit come from the selling of enslaved children, or having the enslaved children grow up on the plantation and be used in the fields or in the home, it truly depended on the person, the slavemaster, and the plantation. One person who had no control over that outcome: the mother.

Edit: formatting

More of course can be said, but I cover much of this in this older answer of mine.