Pre-1960s scholarship appeared to think so, and certain archaeological discoveries seemed to support this, as parallels to events in the text were found in documents from the 2nd millennium BC at Nuzi. From the late 60s, early 70s there was a general reaction against this, as it was pointed out that those parallels also existed in the 1st millennium, and as a result Abram/Abraham got relegated to folklore status in many circles.

Scholarship is somewhat divided between those who think that it's possible he existed (and that he fits a 2nd millennium context), and those who think he's a mythic father-figure for the Israelites, created somewhere in during the Babylonian Exile, (along with the rest of the Pentateuch). Lacking sources, it's impossible to tell whose right, as depending on how you look at it, the evidence can support both interpretations.

The question at some level is beyond answering (and some argue, not even worth trying to answer), as the possibilities of finding evidence of a single man if he existed are quite remote. We have little evidence of many kings from the area, so finding an itinerant man is probably pushing it. Most people would argue that everything before the Monarchy period is fraught with historicity problems, but whether you argue that there is nothing there, or whether you argue something is there, is often decided along confessional lines.