According to the Evening Standard, Pulisic is considering cutting short his summer break in order to be in top shape ahead of his debut season in England.
Giroud has only appeared in seven matches across all competitions for the Blues this season. The former Arsenal star finds himself behind Tammy Abraham and Michy Batshuayi in Lampard's pecking order.
Chelsea currently sit 10th in the Premier League and 11 points behind fourth-placed Tottenham, while they must overturn a 1-0 defeat to Borussia Dortmund in order to progress further in the Champions League.
His arrival is expected to replace current world footballer of the year Luka Modric, who has fallen down the pecking order at the Bernabeu this season.
The former Chelsea star has found himself out of the Red Devils' starting XI this season. The Serbia international only played the final 22 minutes in Man United's Premier League 1-1 draw against Southampton last weekend.
Pepe has made just five Premier League starts so far this season. The likes of Emile Smith Rowe and Bukayo Saka are ahead of him in Arteta's pecking order.
Henderson started for Man United in their 2-1 win over Brighton on Sunday, but it is believed problems between De Gea and Solskjaer go much further back.
Arsenal boss, Mikel Arteta, wants the club to sign three players this summer transfer window in order to fit his plans to play a 4-3-3 system next season.