It was feared that Enright would miss the rest of the season after he suffered a medial ligament strain in the Cats' qualifying final loss to Fremantle.
But the five-time All Australian was able to complete a solid running session on Tuesday morning, which was overseen by club physio Mike Snelling.
"Corey is going pretty well," Snelling told CatsTV on Tuesday afternoon.
"We were able to take him outside and run him around today, which was good. It was a step up.
"He's obviously still got some marks to pass before he's back in training or he's passed fit for the weekend.
"So we've got some criteria that we've set for him, and we'll just have to see how he goes for this week."
There is even better news for another Geelong defender, with Jared Rivers set to take on the Hawks after recovering well from the ankle injury that led to him being subbed out of the Cats' win over Port Adelaide last weekend.
The former Demon was injured when his leg became caught under Power forward Jay Schulz, who was trying to pick up a bouncing ball.
"He was pretty fortunate to be honest," Snelling explained. "It looked pretty horrible. He could've done some pretty severe damage.
"He did come off the ground, and obviously didn't take any further part in the game … but he's come up really, really well.
According to Snelling, Rivers will take part in Geelong's main training session at Simonds Stadium on Wednesday morning and "he'll be available for selection, hopefully, for the weekend".
Twitter: @AFL_AdamMcNicol
