It depends on what enslavers we're talking about and when. Louisiana sugar planters were famously devoted Whigs. Zachary Taylor, one of only two Whigs to win the presidency, was one of them. Throughout the South, for as long as their is a functional two-party system, the Whigs and Democrats are usually competitive. Whigs tend to do somewhat better in the Upper South (the non-cotton slave states), Georgia, and the aforementioned Louisiana. Democrats do better elsewhere. There's also sometimes tendency for the extremely wealthy enslavers to be somewhat more Whiggish than the merely rich.

But that only tells us about a few decades, roughly 1828-1854. Before that, the United States doesn't have political parties in the modern sense. There are groups that sometimes function that way, but they're more fluid and personality-oriented than they will be subsequently. With a few exceptions, the South before the two-party system is strongly in favor of the Republican party...but not our Republicans. These are the Thomas Jefferson and James Madison Republicans, who positioned themselves as the champions of democracy (except for slaves) against aristocracy (except the enslaving kind). The Federalists had prominent individuals in Southern politics, most notably Washington, but had trouble sustaining a movement in the section.

Afterwards, things get simpler. Between roughly 1852 and 1854, the southern wing of the Whig Party collapses. They were not strong enough to restrain their increasingly antislavery northern wing and took the consequent beating. They just couldn't deliver for slavery as well as the Democrats could, which the Dems demonstrated by mostly successfully forcing their own antislavery wing into submission during the battles over the Mexican War and Compromise of 1850.

At least initially, that manifests more in Whig voters staying home. Many will go over to the Democrats in time, but not before trying several other alternatives over the course of the 1850s. For many, that's the anti-immigrant Know-Nothing party. For others it's just running as individuals or under vague platforms espousing Union. Some just quit politics. That leaves the South as an avowedly Democratic section, at least on a functional level. It's a significant change, but the Democracy was a party with a coalition centered on the South (it's most united, disciplined member and consequently of outsize influence) from its beginning. That only changes in the twentieth century, which is beyond my expertise.

Sources

Cooper's Liberty and Slavery gives a high-level overview of partisan alignments in the antebellum South. It's also a pretty fun read for a history book.

Freehling's The Road to Disunion (2 vols) is a much more detailed approach.

Howe's The Political Culture of the American Whigs doesn't address Southern Whiggery very much, but uses Alexander Stephens' (future Confederate VP) life to trace the path many ex-Whigs in the section took through the 1850s. His What Hath God Wrought is a disgustingly good account of the Second Party System's rise and high tide, to the point where it's a profoundly disappointing that he didn't go on a few more years and get into the complete collapse.