Sky Sports said an agreement has been reached between the clubs, who are both competing at the Club World Cup in the United States.
The Blues will reportedly pay up to 65 million euros (£55 million, $76 million) for the 20-year-old, who had been in Chelsea's academy as a youth player.
Gittens eventually moved to Manchester City's academy before signing for Dortmund in 2022.
He made 107 appearances for the German giants, scoring 17 goals, 12 of which came in the Bundesliga and Champions League last season.
Chelsea were in the market to bolster their wide options after refusing the chance to sign Manchester United's Jadon Sancho on a permanent deal after his loan spell at Stamford Bridge.
Mykhailo Mudryk also remains provisionally suspended by the English Football Association after failing a drugs test.
Gittens would be Chelsea's fourth signing of the summer, joining striker Liam Delap, midfielder Dario Essugo and defender Mamadou Sarr.
Brazilian rising star Estevao Willian, 18, will also join the Premier League giants after his Club World Cup campaign with Palmeiras.
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