Gibson was unable to stop Boro from suffering relegation to the Championship last season but his impressive performances throughout the campaign had led to suggestions he would be picked up by a top flight club this summer.
West Brom have been leading the chase for the former England Under-21 international but there has been no progress made over any potential deal since the transfer window opened.
The Baggies are believed to have had two offers rejected, the highest said to be worth around £20million, and the club are ready to return with an improved bid in order to land the centre-back.
Boro manager Garry Monk is desperate to hold onto the 24-year-old as the former Leeds boss aims for automatic promotion but he may not be able to stop a departure.
An offer around the £25m mark is likely to be enough to see West Brom land Gibson.