Holders Celtic face a short journey to take on St Mirren in the fifth round of the Scottish Gas Scottish Cup.
Rangers have been handed a home draw against Ayr with only one all-Premiership tie in the last 16.
Aberdeen are at home to Bonnyrigg Rose, who knocked out Falkirk, while Airdrie have been rewarded for their victory over St Johnstone with another home tie against Hearts.
Hibernian face a journey to Inverness to face Duncan Ferguson’s team while Motherwell are travelling to Greenock to take on Dougie Imrie’s Morton side.
Partick Thistle have the chance to knock out another top-flight club following their win over Ross County, after being drawn at home to Livingston.
Kilmarnock will find out their opponents on Tuesday when Brora Rangers take on Cove Rangers for the right to travel to Rugby Park.
The games will take place on the weekend of February 10-11.
Scottish Gas Scottish Cup fifth-round draw
- Kilmarnock vs Brora Rangers or Cove Rangers
- Inverness vs Hibernian
- Aberdeen vs Bonnyrigg Rose
- Morton vs Motherwell
- Airdrie vs Hearts
- Rangers vs Ayr
- Partick Thistle vs Livingston
- St Mirren vs Celtic
Rodgers: Vata has to earn his chance at Celtic
Rocco Vata celebrates with Daniel Kelly
Celtic manager Brendan Rodgers told Rocco Vata he had to continue earning the right to play after the 18-year-old scored his first goal in a 5-0 Scottish Gas Scottish Cup victory over Buckie Thistle.
The winger has been linked with clubs such as Sampdoria, Bologna and Como and his contract expires in the summer, but he was handed his first appearance of the season and tapped home from close range to round off the win.
Rodgers, who also brought 18-year-old debutant Daniel Kelly off the bench, said: “Young players have to earn the right. There’s been a lot of noise around Rocco but any young player has to earn the opportunity. I’ve given many young players opportunities in my career.
“But I like Rocco, he has qualities. He is strong, he’s aggressive, he wants to get goals. When he came on he got his goal and he had other opportunities.
“And Daniel Kelly is a young player I really like. If he keeps progressing and developing he will have a big future. He is left-sided, he is quick and strong, he presses the game very well.
“They get a taste of it and hopefully that gives them the motivation and determination to continue progressing.”