The France international has been frustrated at his limited game time with the club this past season, having found himself behind Alexis Sanchez in the pecking order for a majority of the season with just 11 league starts.
Having also featured as a substitute in the FA Cup final with Danny Welbeck preferred ahead of him, Giroud recently admitted that he would consider leaving the Gunners in search of more game time.
"I have had limited playing time for several reasons but that is not to say that I lost the confidence of the coach," he said. "I will not be content with limited playing time next season."
"I will have to think carefully to consider it with my relatives and my advisers. It [a transfer] will be a well-thought out decision. But, yes, I want more playing time."
With manager Arsene Wenger reportedly meeting Lyon president Jean-Michael Aulas in France to discuss a possible move for Alexandre Lacazette, there was reported interest in Giroud from the likes of Marseille, Lyon, Inter Milan and Napoli.
However, West Ham are now also interested in the former Montpellier man and could match a reported £20m ($25.5m) offer to keep him in London according to The Mirror.
Having struggled for goals last season with no player scoring double-digit figures, Hammers boss Slaven Bilic is looking to add some firepower to his faltering front-line.
Giroud – who renewed his contract with the north London club in January – has been with Arsenal since a £9.6m ($12.2m) move from Montpeller in 2012 and has gone on to score 98 goals in 226 games in all competitions.