Arsenal and Manchester City are also reportedly monitoring the 30 year old, but the Baggies are yet to receive a bid from either club for the Northern Irish international.
With no movement from other teams, the Foxes are looking to take advantage of the situation by making the first move for Evans.
Like the Gunners and City, Leicester also made a bid for the centre back in the summer, but all offers for the player were rejected by West Brom at the time.
With Evans now keen to depart the Hawthorns, and refusing to sign a new deal which currently has 18 months left to run, the Baggies are more open to a deal happening in January. They are said to be looking for £25m to finance a move for a striker.
Whether a bidding war will ensue if Leicester do make a bid, it remains to be seen. Considering Arsenal’s defensive problems this season, you would have thought recruits at the back would have been a priority for Arsene Wenger, but there has been no urgency this month to sign a new centre half; whether that is Evans or someone else.