Giroud's Chelsea contract expires at the end of the season, and Pellegrini is keen to "add quality to his front line," per the report.
Giroud only arrived at Stamford Bridge from Arsenal in January. He managed to force his way into the team under Antonio Conte and ended the season as first choice ahead of summer signing Alvaro Morata.
He also helped the Blues win the 2018 FA Cup final with a 1-0 win over Manchester United at Wembley Stadium. Giroud subsequently went on to play a key role for France as Les Bleus won the FIFA World Cup in Russia.
He was linked with a move away from Chelsea over the summer. Marseille were interested in bringing him back to France, but Giroud said he did not want to leave, per Joe Krishnan at the Evening Standard.
Giroud returned to Chelsea late after his World Cup exploits and has made seven appearances in the Premier League but is yet to find the back of the net.
He has still proven to be effective:
Morata has also struggled for goals and has just three in all competitions. Sarri has explained why he is not worried by his misfiring strikers:
Nizaar Kinsella @NizaarKinsella
Sarri on playing a false 9 instead of Morata and GIroud: "I want to use in this moment Giroud-Morata as I think it is important in this part of the seaosn. Giroud is important for his team-mates and I think today, Morata was more aggressive. I am not worried." #CFC #LFC #CHELIV
Giroud has been an important signing for Chelsea. He is excellent in the air, holds the ball up well and gives the Blues a different dimension in attack.
If he were to leave midway through the season it would leave Chelsea short of striking options. The club have allowed Michy Batshuayi and Tammy Abraham to depart on loan which would leave just Morata and 17-year-old Callum Hudson-Odoi.
Under Sarri the Blues are undefeated in the Premier League and level on points with leaders Manchester City. Meanwhile, West Ham are four just points off the bottom after a tough start to the season.
Giroud may not be a guaranteed starter at Chelsea, but he remains an important figure. The club look to have an exciting future under Sarri, and it's unlikely Giroud would want to swap that for a move to West Ham, even if he could expect more game time.