The France international only made 14 starts in the Premier League and Champions League last season, although he still weighed in with 14 goals for the Gunners.
Walcott, meanwhile, started 23 of Arsenal's 38 league fixtures last term, but a switch to a 3-5-2 formation in the latter stages of the season saw him dropped to the bench.
According to the Daily Star, the Hammers are eager to recruit Giroud and Walcott to strengthen their squad and improve on a below-par 2016-17 campaign.
The report also claims that Gunners boss Arsene Wenger would not stand in the duo's way should they opt for a move to the London Stadium.
Giroud is also thought to be on Lyon's radar, with the Ligue 1 club trying to recruit the former Montpellier HSC marksman as a replacement for Alexandre Lacazette.