Sky Sports News are reporting that two offers for the 'Boro centre-back have been turned down, the first of approximately £15m, before a follow-up bid totalling in the region of £20m was also refused.
Both bids fall short of Boro's value of the player, who is related to Middlesbrough chairman Steve Gibson.
Future England defender
As part of former boss Aitor Karanka's side that earned temporary promotion back to the Premier League last season, Gibson has earned plaudits for his leadership qualities and aerial prowess in defence.
Early last term, Gibson was linked with a switch to eventual league champions Chelsea to replace the departing and ageing John Terry at the heart of the Blues defence, before interest cooled.
In March, the 24 year-old earned his first senior England call-up replacing the injured Chris Smalling. Despite having yet to receive his first full cap for the Three Lions, many are touting Gibson as future England regular.
Pulis seeks further recruits
Defence has always be a element that Baggies boss Tony Pulis has emphasised more strongly that any other part of his team's setup.
With Jonas Olsson having been phased out of team last season and with Gareth McAuley in the final years of his career, Pulis sees Gibson as the ideal candidate for a place in his back-line, as a lofty centre-back with also an eye for goal.
Following the addition of 6 ft 5 inch defender Ahmed Hegazy, brought in on loan from Egyptian Premier League side Al Ahly, the recruitment of Ben Gibson also would hand West Brom a number of choices in defence this term.
It remains to be seen if the Baggies return with another bid for the defender.