The Constitution addresses slavery directly three times, though in all three the founders elected to obscure their doing so. All three are occasions where they go out of their way to grant slaveholders extra security in the new government. These are:

The 3/5 Clause, which counted slaves as 3/5 of a white person for purposes of granting seats in the House of Representatives and electoral college, rather than 0/5 as would be the case for other people considered outside the American body politic, like Native Americans. Furthermore, those seats aren't going to represent the slaves, but rather their enslavers. So this literally lets slaveholders buy themselves extra representation in government and consequently disproportionate power.

This is sometimes presented as a compromise, but that's in error. Much later on, a possible liability in direct taxes was apportioned the same way but there was no such quid pro quo at the time that 3/5 was agreed to and any future direct taxes would require the votes of the House...which was built with the 3/5 ratio.

The Framers granted the Congress the power to pass navigation acts (regulation of imports and exports) by a simple majority like everything else. There were proposals to subject them to a supermajority otherwise, largely hailing from the Lower South. But there's an exception to that power in the slave trade clause, which made it acceptable to them: the Congress could not, under any circumstances including constitutional amendment specifically for that purpose, prohibit the importation of slaves until 1808 at the earliest. Only then does Congress gain the ability, and only prospectively. It's not required that they set up an import ban. As an adjunct to this, Congress does gain the power to lay a duty on slave imports but it's specifically capped. This option is never used.

The result isn't that every state imports slaves; they have the option to do so and some demur. North Carolina, Georgia, and South Carolina do not and bring tens of thousands of slaves into the country during that window.

Back on the domestic front, we have the Fugitive Slave Clause. It used to work out that if you had a slave who stole themselves, on the premise that they owned their own lives, they might run to another state and you might have trouble getting them back. States were not required to oblige you in recovery. This clause grants a special right of extraterritoriality to slavery by giving enslavers that right to recover their human property across state lines. It's initially unclear how that will operate, but Congress comes up with a fugitive slave act in 1793 that provides basic regulations and some obligation by the national government to take an active role. They manage to beat back a set of recommendations from the South for more stringent provisions then, but a law largely on those specifications is later passed in 1850 and prompts a wave of massive resistance.

I wouldn't construe any of that as including African-Americans as the term is usually meant, but it is true that the Constitution is silent on things like the requirements of citizenship. There's no explicit mention of race in it at all, though the three clauses above make it functionally proslavery. It's true that some states permit black Americans who meet property requirements to vote in elections, including those held pursuant to ratification of the Constitution. Those rights are steadily eroded just as white suffrage expands.

However, the question is narrowly construed. If we take a slightly broader look at things we can get a better answer as to what kind of polity the framers looked forward to. The First Congress passed a Naturalization Act in 1790 which set out the rules for becoming an American. It begins, after the standard enacting line:

That any alien, being a free white person, who shall have resided within the limits and jurisdiction of the United States for a term of two years, may be admitted to become a citizen thereof

I don't think they could have written a clearer statement that their country was for whites only and everyone else was just sort of living there. That said, the status of black Americans already in the country remains ambiguous under federal law until Roger Taney rules on the matter in 1857. Then, he declares them non-citizens, incapable of citizenship, and possessed of no rights a white man was bound to respect. That was fully reversed only by the Fourteenth Amendment in 1868.

Sources

David Waldstreicher's Slavery's Constitution is my go-to book on the subject. Kaminski's A Necessary Evil? collects primary sources from the debates at the convention and during ratification that address slavery.

Works of a larger scope include Don Fehrenbacher's The Slaveholding Republic, which is good on the facts, but works too hard trying to excuse the framers for the house they purposefully built even as it traces every board in it all through the Antebellum. More in line with Waldstreicher are Mason's Slavery and Politics in the Early American Republic, his edited volume with John Craig Hammond Contesting Slavery, and Finkelman's Slavery and the Founders. Finkelman is especially good. My position on the 3/5 provision follows him.