There is nothing special about borders of former Russian Empire when considering Soviet expansion.
There is an official USSR diplomatic note to Germany (Molotov to Schulenberg on 25 November 1940). It is a conditional agreement of USSR to join Axis. The conditions were to extend USSR sphere of influence to: Bulgaria, Bosfor, Dardanelles, and south of Batumi and Baku towards Gulf of Persia (source: Mark Solonin "25 June" p. 227, citing the Russian Presidential Archive set 3/64, doc 675 p. 108). These territories were obviously not in the former Russian Empire. Molotov sent a note acting on Stalin's orders.
"Did Stalin want..." What person X wanted or what person Y was thinking is not a good subject of historical research; our motives and desires are mysterious even to our close ones, so how could historians seventy years later know better?
What we know is that USSR did not hide from Germany their interest in expanding to these territories, and I think this is what your question is about, right?
Also, in 1945, Stalin negotiated with Allies to officially take a part of German Prussia (Koenigsberg, now Kaliningrad) - it was never a part of a former Russian Empire. (Unofficially, Stalin had much control over East Germany, Czechoslovakia, Hungary, Bulgaria, etc. These were never parts of a former Russian Empire.)