The 32-year-old has featured just seven times since he arrived at The Hawthorns as a deadline day free transfer last August.
Kelly has started just four league games, and not since mid-October, for Albion. He was a target of new Latics boss Shaun Maloney and the experienced former Liverpool defender has been allowed out.
The defender started both Chesterfield FA Cup ties under Carlos Corberan but his only league involvement has been a few minutes at Sunderland after the World Cup break.
He is contracted at The Hawthorns until 2024 after penning a two-year deal last September.