The Baggies are already thought to have seen three offers rejected for the 24-year-old England international, the most recent of which totalled a club record £21million including add-ons.
But they have not ruled out making another approach with boss Tony Pulis eager to reinforce his defence ahead of the Premier League campaign.
A deal for Bournemouth’s Marc Wilson appears likely with Cherries boss Eddie Howe admitting the centre-back, who spent the second half of last season on loan at the Hawthorns, is likely to leave this summer.
Manchester United pair Chris Smalling and Phil Jones also remain on the radar, along with Tottenham’s Kevin Wimmer.
Yet Gibson is to this point the only one subject to a firm offer, with the Baggies meeting stern resistance from Middlesbrough on each occasion.
Reports yesterday claimed the Championship club, who were relegated from the Premier League last season, would be prepared to offer Gibson an improved contract in a bid to stave off any interest once and for all.
Boro are in any case keen to make clear they are under no pressure to sell a player who featured in every single league game last season and made his senior England bow in March.
They are thought to value Gibson at a similar price to his international counterpart Michael Keane, who moved from Burnley to Everton earlier this summer for a deal worth an initial £25million.
Striker Jay Rodriguez and defender Ahmed Hegazi remain the Baggies only signings of the summer so far.
Reports yesterday meanwhile claimed playmaker Nacer Chadli is a possible target for Swansea this summer, should they sell Gylfi Sigurdsson.
Chadli did not travel to Albion’s training camp in Austria following a disagreement with Pulis over his fitness regime but did feature against Leicester and Crystal Palace in the Premier League Asia Trophy.
Former Albion forward Izzy Brown has meanwhile joined Brighton on a season-long loan from Chelsea, after signing a four-year deal with the Premier League champions.