The Blues switched their focus to Giroud and Tottenham's Fernando Llorente after failing in their attempts to sign Edin Dzeko from Roma. Giroud travelled with Arsenal for Tuesday night's match against Swansea City, but the 31-year-old will likely be left out ahead of the shock move to the rival club.
Chelsea have been able to convince Arsenal to sell Giroud to them with Dortmund needing a replacement before sanctioning the sale of Aubameyang. Dortmund have earmarked Michy Batshuayi, who will now be able to make a loan move to Germany to replace the Gabon international.
Goal understands Dortmund will pay a €1.5 million loan fee to secure Batshuayi's services until the end of the season.
Batshuayi's influence has improved Chelsea's leverage in securing the France international striker, who won't arrive in time to face Bournemouth but could be available for Chelsea's trip to Watford next Monday night.
Giroud currently has 18 months remaining on his Arsenal contract and Chelsea are expected to break their club policy of only giving over-30 players one-year deals.
Chelsea will also bring in Emerson Palmieri on transfer deadline day from Roma in a deal worth £17.5m with a further £8m in add-ons.