Not quite what you're asking, but there is a link between the Romans and the mammoths - that megafaunal remains in Renaissance Italy were regularly and for a while 'officially' misidentified as the bodies of the elephants that Hannibal led across the Alps. I wrote an answer on it here., which remains the only time I've had an excuse to write 'inordinate fecal deposition' on AH.

There is a book on this subject by Adrienne Mayor: The First Fossil Hunters: Dinosaurs, Mammoths and Myth in Greek and Roman Times. Essentially, the Greeks and Romans fairly regularly dug up extremely large bones, at least some of which probably belonged to ancient megafauna, but never really made the connection. Instead, they were generally rationalised as belonging to what we would think of as mythical creatures, such as centaurs, giants and other monsters. Mayor argues, for instance, that the legend of the griffin came from Scythian gold miners, who encountered the remains of creatures like Protoceratops in the Gobi desert.

Mayor's book hasn't attracted much attention from classicists (as far as I can tell, it wasn't reviewed in any of the major Classical journals) and I'd generally be inclined to be a little sceptical of her attempts to identify specific extinct animals being misidentified in the ancient sources (our sources weren't exactly hung up on paleontological detail - whether some of these finds were anything more than big bones or even someone hearing that someone else had found some big bones is debatable). With that said, the two key points of her thesis are sound. Firstly, the Greeks and Romans did accept the concept of 'extinction' - though they didn't use that word, they did believe that the stories of semi-divine heroes and (most) monsters belonged to a lost world that no longer resembled the one they lived in. Secondly, she is right that Greek and Roman stories of oversized bones are invariably link them back to that lost world, rather than positing that they were ancestors of familiar animals. So no, unfortunately, they never knew about wooly mammoths, even if they may have found some of their bones.