With the team struggling to get in the win column and Barrie failing to produce, the Leafs have received calls regarding the blue-liner, according to Sportsnet's Elliotte Friedman.
"I'm hearing Toronto's received calls on him," Friedman said on Hockey Night In Canada's "Headlines" segment. "So far, they've rejected the idea, they don't want to trade him."
The Leafs sent Nazem Kadri, Calle Rosen, and a 2020 third-round pick to the Colorado Avalanche for Barrie, Alexander Kerfoot, and a 2020 sixth-round pick. Colorado also retained half of Barrie's salary in the trade.
Barrie is set to hit free agency at the end of the season. He currently carries a cap hit of $2.75 million.
"He has not asked for a trade. However, I think that there's an understanding that it hasn't worked," Friedman added.
Barrie has yet to score a goal this season and has six assists in 22 games. He recorded 14 goals and 59 points in 2018-19.