The 30-year-old has become the latest defender to be sidelined due to injury.
Scans have shown that the 30-year-old has suffered a hamstring strain, meaning that he will join Dedryck Boyata and Jozo Simunovic on the sidelines, leaving Ronny Deila with a problem at the heart of the defence for the game against Robbie Neilson's men.
While Erik Sviatchenko has been impressive since his January move, Mulgrew's injury may see Deila turn to Efe Ambrose for Saturday's game.
Ambrose has made just three appearances since being sent off in the early stages of the Bhoys' League Cup semi-final defeat to Ross County in January, so may not be in the best form for the clash at Tynecastle.
Luckily for Deila, while The Scottish Sun claims that Mulgrew could be out for up to three weeks, the report suggests that he is hoping to be back for the Hoops' potentially crucial game against Aberdeen the week after they face Hearts.
Celtic are currently nine points clear at the top of the table with four games to go, so could clinch their fifth league crown in a row if they beat the Jambos and the Dons fail to beat Motherwell.
But should Derek McInnes' men win their game, the two sides will meet at Parkhead the following Sunday knowing that anything other than an Aberdeen victory will confirm Deila's men as champions.