The full-back returned to action as a substitute in Tuesday's FA Cup third-round replay defeat to QPR having suffered the injury at The Emirates on 8 December.
Albion had not disclosed the full extent of Reid's injury, but the Baggies boss has now admitted that Reid played on despite sustaining a fracture.
Clarke said: "We didn't release the injury but it was a fractured fibula he got against Arsenal. He played for 45 minutes with a broken leg, which shows the character of the boy.
"I've put it out there because I want people to see the character of the boy.
"Sometimes people think 'Steven is always injured' but he has an incredible strong mental strength and he wants to play."
West Brom face local rivals Aston Villa at The Hawthorns on Saturday, and Clarke is considering a recall for Claudio Yacob.
The midfielder has been sidelined for six weeks with a hamstring injury, but he has returned to training and has an outside chance of making the squad against Villa.
Clarke added: "The game might come too soon, but we will have a look at him and make a decision. He's in a good place."