After spells at Shanghai Shenhua and Juventus, the former Chelsea striker returned to England in the summer.
However, the death of his agent, Eric Manasse, reportedly made Anelka consider retiring before he decided to continue playing.
Clarke told the club's official website: "Nicolas is happy to be here and he's delighted to try and help this team.
"He wants to finish his career well. When he was in China he didn't want his career to fizzle out. He didn't want to finish the season sitting on the bench for Juventus.
"He wanted to come back to the Premier League, show everyone he's a good player and finish his career on a high."
Anelka could return for West Brom's match against Swansea City today.