In essence, no; there was no mechanism in law whereby an enslaved person could inherit the property of their master. Slaves were property themselves first and foremost and did not have the necessary legal identity to inherit (or even really own in general) property; furthermore, slave status was determined maternally rather than paternally. As such, any child born to a slave woman and a white man was considered a slave and could in no way, shape or form be a legitimate heir unless first freed by their owner; whilst a child born of a free mother would not be a slave.

There were court cases that confirmed and upheld this state of affairs. In 1792, a slave owner in North Carolina tried to bequeath part of his plantation to a slave woman he had owned; after his death, the will was reviewed by a court in 1801, which found that the mandate was invalid because:

slaves have not that competence; for a civil incapacity results from the nature and condition of slavery. And it would be a solecism, that the law should sanction or permit the acquisition of property by those whom it afterwards withholds that protection without which property is useless [...] for a slave can bring no action; he can neither acquire nor transfer property, by descent or purchase; nor will perscription avail him, to assert a title against his master [source]

This is standard for slave law throughout the antebellum South: slaves do not have the right to own or inherit property as far as the law is concerned because they are property themselves (and so anything they own is usually the property of their master; anything they use is by his consent), and the legal system does not recognise them as persons proper, preventing them from being able to meaningfully enforce any claim to property right.