Rose could face Bathgate
But Rose will first have to get through today's replay at Prestonfield against Ayr side Whitletts Victoria.
If they do, a large crowd can be expected at Creamery Park for the clash of the two East Junior giants next Saturday.
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Hide AdFifers Hill of Beath Hawthorn have a fifth round tie at home to Shotts to take care of before they can start to think about playing the beaten finalists from last season, Clydebank.
Lochee were drawn at home to Tayport in a repeat of the 2004/5 final and Cambuslang or Rutherglen or Irvine Meadow were drawn against Largs.
Quarter-final draw (ties to be played on Saturday 13 March):
Bathgate v Linlithgow or Whitletts
Lochee United v Tayport
Cambuslang or Rutherglen or Irvine Meadow v Largs
Clydebank v Hill Of Beath or Shotts Bon Accord