The bid is believed to be Tony Pulis' side's third this summer but the 'Baggies' have been told in no uncertain terms that Gibson will not be leaving.
The offer was worth an initial £18m with a further £3m in add-ons but Boro are reluctant to part with the defender who came through the club's academy and served the club with distinction as an ever-present last season.
Gibson, who is the nephew of boro owner Steve Gibson, was rewarded with an England call-up for the recent games against Scotland and France as a result of his fine form last campaign.
Nevertheless, the Teessiders are keen to keep hold of the Nunthorpe-born centre-back as they look for an immediate return to the Premier League.
Gibson started Middlesbrough's latest pre-season friendly against Mansfield on Saturday, but was withdrawn after just 15 minutes with a suspected broken nose.