At the same time, Babcock said recent visits to the homes of Matthews in Arizona and goalie Frederik Andersen in Denmark were outstanding and have helped the players understand where the coach is coming from.
And vice versa.
"Over [Auston's] career, over the time when I'm coaching him, there are going to be lots of times he's not happy," Babcock told NHL.com from the 2018 IIHF World Hockey Championship in Denmark on Monday. "That's life.
"When he wants to be pushed, he wants to be the best player he can be. That's what I'm going to do, for sure. We're always going to continue to have dialogue. That's the same for Freddie, that's the same for any of them. They want to get better, I want them to get better. We're all in it together.
"Does that mean things are going to be rosy every day? No."
A day after the Maple Leafs were eliminated in seven games by the Boston Bruins in the Eastern Conference First Round on April 25, Sportsnet reported that a rift had formed between Babcock, 55, and Matthews, 20. Matthews, who had 63 points (34 goals, 29 assists) in the regular season, had one goal and one assist against the Bruins. He played six shifts -- a total of 4:05 -- in the second period of the 7-4 loss in Game 7 and looked unhappy on the bench in the third, when he played 8:16 and Toronto allowed four goals.
Babcock and Matthews insisted there were no issues between them after they had their end-of-the-season meeting at Air Canada Centre on April 27.
Video: BOS@TOR, Gm3: Matthews goes top shelf on Rask
It's a stance Babcock maintains.
"Way too much has been made out of this, I can tell you right now," he said. "But right now, that doesn't matter. What matters is, we continue to get better."
Such improvement isn't limited to on-ice performance.
Strides also must be made in communication, which Babcock is attempting to do this offseason. Along with meeting with Matthews and Andersen at their homes, he had sit-downs with defenseman Nikita Zaitsev and forward Kasperi Kapanen at Worlds.
Babcock's assessment of his talks with those four players: "I just think outstanding."
He was especially encouraged by his chats with Matthews and Andersen, 28, who set a Toronto record with 38 wins in the regular season, but struggled in the first-round loss, especially in Game 7, when he allowed six goals on 35 shots.
"It's a different kind of special when you get to go to a guy's home because you learn a lot about the guy, from the pictures on the wall, his dad's office, learning about their homes, how things are … it's better," Babcock said.
"I thought they were fantastic. Both guys are huge parts of our team. All four guys, actually, are huge parts of our team, and we need them to continue to get better. These are all young guys who have an opportunity to have a huge impact on our franchise."
What Babcock also has to deal with is the future of the Maple Leafs front office.
Kyle Dubas was promoted to general manager from assistant GM by team president Brendan Shanahan on Friday. It remains to be seen whether director of player personnel Mark Hunter, who was in the running for the GM job, and/or Lou Lamoriello, who was reassigned to adviser after a three-year reign as GM, stay with the organization.
Babcock would love to see them remain together.
"I just think, I was hired by [Shanahan], [Hunter] and Dubas, and Lou wasn't there yet," Babcock said. "And so we brought Lou on to help us all get better. I'm still hopeful Lou is going to stay on. I'm not sure he is going to.
"My biggest thing is, we're better together than we are apart. The more quality people you have, the better off you are. Let's find good roles for everybody to make everybody important and let's continue with our program so we can continue to get better."
Video: BOS@TOR, Gm6: Andersen flashes the leather for save
It has been a bittersweet offseason for Babcock. Though he has been encouraged by his talks with Matthews, Andersen, Zaitsev and Kapanen, he is mourning the loss of former University of Alberta coach Clare Drake, a mentor to Babcock who was inducted into the Hockey Hall of Fame on Nov. 13, 2017.
Drake died at the age of 89 on Sunday.
"He'll be missed, but we'll be laughing about his stories for a long time," Babcock said. "He cared about you and made you a better man.
"Everybody talks about him as an innovator and about his technical ability but his ability to deal with people definitely was his best asset."
Babcock is attempting to do the same by visiting with some of his key performers. Though his no-nonsense approach can sometimes rub players the wrong way, at least he's making an effort.