The French striker, targeted by Everton and Marseille among others, is understood to be keen on a move away from the Emirates after Arsenal signed Lacazette for a club-record £52 million.
West Ham are in the market to sign at least one striker, and were interested in signing Giroud until they were told East London is not the Frenchman's first-choice destination. They will look into other targets such as former United striker Hernandez or Villarreal man Bakambu.
The Hammers' hopes will be lifted, however, as they close in on a deal to bring England international goalkeeper Joe Hart. The 30-year-old is expected to leave Manchester City this summer, and West Ham are stepping up the efforts to secure his services.
Hart wants to stay in the Premier League, and he could join Bilic's side on loan before making the move permanent next summer.