The club are hoping to sign him before Tuesday’s deadline as they want him to play in the Club World Cup.
Dortmund are also taking part in the competition.
The west London club are looking to strengthen in the winger position following the decision not to take up the option to sign Jadon Sancho permanently.
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The 20-year-old winger was a target in January and has remained high on a list of eight or nine attacking names Chelsea are working on this summer.
Gittens is valued at £50m but lost his place in Dorrmund’s starting XI towards the end of the season.
Chelsea may also sign another striker after already completing deal for Liam Delap with Hugo Ekitike and Benjamin Sesko on their shortlist.
