The center forward has emerged as a key target for manager Julen Lopetegui.
Per Sky Germany, West Ham are ready to offer the 31-year-old a there-year contract.
Fullkrug, who was at Euro 2024 with his nation, is available for £26.2 million.
However, the Hammers have yet to reach that mark, as they have bid as high as £21 million.
The deal does include incentives, which the club may have to raise to get Dortmund to accept their offer.