The Gers are due to play their city rivals Queen's - bossed by former Light Blues defender Gus MacPherson - at Airdrie's Excelsior Stadium on Tuesday, August 26, in a re-arranged first-round match.
Whoever wins that game will go forward to play Highland club Caley Thistle at a date that has still to be decided.
The last time Ally McCoist's side squared up to their former top-flight opponents was two seasons ago in the same competition when they crashed to a 3-0 loss at home.
Elsewhere in the second round, Partick Thistle will travel to face League One Morton, while St Mirren will take on Dunfermline in Paisley.
Hamilton will welcome Alloa Athletic to New Douglas Park, with Ian Murray taking his Dumbarton side to one of his former clubs, Hibernian.
Kilmarnock will play local rivals Ayr United at Rugby Park.
Celtic will enter the competition along with defending champions Aberdeen, Motherwell, Dundee United and St Johnstone in the third round.
Hibernian v Dumbarton
Greenock Morton v Partick Thistle
Falkirk v Cowdenbeath
Queen's Park or Rangers v Inverness Caledonian Thistle
Hamilton Academical v Alloa Athletic
Stranraer v Ross County
Dundee v Raith Rovers
Livingston v Queen of the South
Kilmarnock v Ayr United
Stenhousemuir v Heart of Midlothian
St Mirren v Dunfermline Athletic
Ties are due on August 26 or 27 except Queen's Park or Rangers v Inverness Caley