Brown's performances in the youth ranks are thought to have attracted interest from Arsenal and Chelsea, but the 16-year-old midfielder is yet to play for the first-team, despite featuring on the bench a few times.
His first appearance among the substitutes came at Stamford Bridge last month, when he could have become the youngest-ever Premier League player, but Clarke will not rush his progress and also resisted the temptation to play him at Arsenal last weekend.
The youngster continues to impress in training and Clarke is keen for him to gain more experience in the Baggies squad.
When asked about the Arsenal match, Clarke told the Birmingham Mail: "I think the way the game evolved it wasn't the type of match to bring him in.
"Don't forget he's not even 16 and a half yet. He's a very young 16, and is just out of nappies to be honest. But it's good to keep him involved.
"He trained last week, he was excellent and deserved his spot on the bench. It's a great experience for him."