Scottish Cup holders Celtic have drawn Hibernian in the quarter-finals, while Aberdeen will face either Kilmarnock or Rangers.
Brendan Rodger's side, who hammered St Johnstone 5-0 in the fifth round on Sunday, will travel to Edinburgh to face Hibernian, while Hearts make the opposite journey to play Partick Thistle at Firhill.
Rangers face a tough trip to face Aberdeen at Pittodrie if they get past Kilmarnock in their Scottish Cup replay on February 20.
There will be an all-Championship tie at Tannadice with Dundee United awaiting the winners of the Highland derby replay between Inverness and Ross County.
Rangers won 4-2 at Aberdeen last week but that was their first victory in four matches against the Dons this season and they were knocked out of the Betfred Cup at the semi-final stage by Derek McInnes' side.
Alfredo Morelos missed that Hampden defeat through suspension and has been sent off in each of the other three meetings.
Morelos will be back from suspension when Rangers host Killie at Ibrox on February 20 following Saturday's goalless draw at Rugby Park.
Celtic have lost their last two trips to Easter Road and drew their previous game in Leith.
There will be another Highland derby next Tuesday after County's Josh Mullin scored a stoppage-time equaliser in a 2-2 draw at Dingwall just before the quarter-final draw.
The last-eight ties will take place around the weekend of March 2/3.
Aberdeen vs Rangers or Kilmarnock
Dundee United vs Inverness or Ross County
Partick vs Hearts
Hibernian vs Celtic.