The clubs are just two of many Premier League teams said to be keen on the 24-year-old, who has been Boro's star performer despite their relegation back down to the Championship.
Middlesbrough are reportedly keen to keep hold of a player who was recently called up to the England squad, but Gibson has recently acknowledged that he needs to remain in the top flight to stay in contention with the Three Lions.
According to The Mirror, both the Foxes and the Baggies are looking to take advantage of his stance, with a bid expected in the near future.
It has been suggested that Gibson may cost over £20m but given Leicester's recent success and West Brom's Chinese takeover, it is a figure that they are likely to be able to meet.
Gibson has made 153 appearances in all competitions for the North-East outfit.