Archaeology in the modern sense, no. But there were antiquarians. Very famous are those of the roman republic like Varro. Even the emperor Claudius seems to have had antiquarian interests. But all in all these were still para historical disciplines, they didn't really make use of archaeological methodologies.

We have a curious reference to the babylonian king nabonidus who actively looked for ancient a rtifacts while rebuilding ancient temples and dating them in archaeological fashion.

I don't know about the middle ages but given that proper archaeology was born in the 19th century I guess there may have been the casual interested antiquarian but that's it.