Reports earlier on Tuesday confirmed that Watford had pulled out of a deal to sign the left-back after failing to agree on personal terms.
West Brom have now swooped in and agreed a deal in the region of £5m with Arsenal.
Gibbs still needs to complete a medical with the Baggies and sign his contract.
The 27-year-old is entering the final year of his Gunners contract and was told he can leave the Emirates earlier this summer.