The Irons would hate to lose him after the 29-year-old has proven himself to be a solid centre-back over the past two years, following a move from Juventus in 2015.
But if there is even a small chance that it is true then manager Slaven Bilic needs to draw up a shortlist of names capable of replacing the Italy defender.
And might the West Ham boss go after Tottenham Hotspur's Kevin Wimmer?
The Sun reported at the weekend that Tottenham are willing to sell the 24-year-old - but not for less than £20 million.
But although the Austrian stalwart is capable of being a £20 million defender, Wimmer, their fourth-choice centre-half who made five Premier League appearances last season, is unlikely to fetch that amount.
Southampton, according to The Mirror, are reluctant to meet those demands and they won't be the only ones. The reality is that it's an unrealistic valuation by Spurs, who will have to lower their asking price if they want to offload him.
If and when they do, West Ham should be ready to pounce.
He might not have had an eventful 2016-17 campaign, with Eric Dier often used in central defence when Toby Alderweireld or Jan Vertonghen weren't available, but Wimmer did look a brilliant centre-back in the second half of the 2015-16 season when he replaced the injured Vertonghen.
Despite being unfancied by Mauricio Pochettino for whatever reason, Wimmer is certainly good enough for the Premier League and if West Ham lose Ogbonna then Bilic should explore the idea of signing the former Koln man. He won't have a £20 million price tag for long.