The Hammers are said to be awaiting a response from the Super Lig side, who are Aziz's hometown team.
The 25-year-old was a reported target for West Ham manager Slaven Bilic during his time at Besiktas, and the Turkish giants will rival them in their pursuit, DHA claims.
Aziz, who has nine caps for Turkey, is widely regarded as one of the division's best centre-backs and is a proficient reader of the game.
And Bursaspor may favour a sale this summer, with Aziz's value set to depreciate as he enters the final 24 months of his contract.
West Ham are currently well-stocked in Aziz's position, with Winston Reid, Angelo Ogbonna, James Collins and James Tomkins all competing for places.
And the quartet could be joined by academy graduate Reece Burke next term, following an impressive spell on loan to Bradford City.
The Hammers move into Stratford's Olympic Stadium this summer and co-chairman David Sullivan revealed earlier this week the club have ambitious plans regarding transfers.