The striker is expected to join Carlisle United after failing to hit the back of the net in 15 appearances for the Tynecastle side.
Esmael Goncalves looks set to follow Stockton out of the exit door. He has been left out of the squad to face Motherwell on Saturday and could leave the club in the coming days.
Krystian Nowak is another who is deemed surplus to requirements and could follow Stockton out of the club.
Manager Craig Levein said: “Cole and Krystian Nowak have both told their agents they are not happy with not getting game-time.
“I couldn’t promise they would get it before the end of the season so I have said if they find something else I’m more than happy to let them go.”
With Stockton gone and Goncalves likely to leave, Levein will be left short on striking options and is still looking to bring in another forward before the end of the window.
“We need to get another striker and that’s what I’m working on just now,” he said. “We might add another midfielder but we will have to see how it all pans out with Krystian.”
Meanwhile, Hearts have agreed contract extensions with four young players.
Andy Irving, who made his debut in the 3-0 win over Hamilton in midweek, has been rewarded with a new deal until 2020.
Daniel Baur and Callumn Morrison have also been handed two-year extensions, while goalkeeper Kelby Mason was handed an additional year until 2019.
Baur, who has featured twice for the first-team this campaign, will join Albion Rovers on a development loan until the end of the season.
Morrison will go to Championship side Brechin having impressed with League Two Stirling Albion in the first half of this season.