The 29-year-old appeared to injure his knee on the hour mark against the Blues and was in tears after the incident as he was substituted off.
Varane's reaction appeared to indicate that his place at the upcoming FIFA World Cup could be in doubt.
If Varane were absent for the Qatar tournament, it would be a huge blow for France, for whom the centre-back played a major part in their 2018 World Cup win.
But despite plenty expressing their sympathy for Varane, Stam felt that his emotions were unnecessary during a time when Manchester United played a crucial game against Chelsea.
The legendary Dutch defender, who was a key member of the treble-winning Manchester United side in 1999, told Optus Sports:
“He was very emotional. The game played on when he fell down and he just kept lying on the pitch basically and after, when he stood up, he was emotional. It looked like he had a problem with his hamstring or whatever but we don’t know yet."
"I don’t know what to say about this, he probably thinks he’s out of the World Cup but I can’t understand being so emotional on the pitch to be honest."
“It’s not only from a centre half’s point of view, you try to imagine yourself in the situation and why is he so emotional and why he is acting like that. There are other things that are worse than this, is what I think.”
Erik ten Hag unsure of the severity of the injury to the Manchester United star
Following the game, Ten Hag was asked how bad the injury to Varane was, to which the Manchester United boss refused to speculate.
Varane has suffered numerous injury problems since moving to Old Trafford last summer, but has established himself as the first-choice defender under the Dutch boss.
After Manchester United's 1-1 draw with Chelsea at Stamford Bridge, Ten Hag told reporters (per The Mirror):
"It is difficult to say how it is now because it is so soon after the game. We have to wait 24 hours and then we will know what it is. We have all seen he came off but with an injury like this we always have to wait a minimum 24 hours and the medical staff can do their job and can set the diagnosis."