The deal could hasten Arsenal's signing of striker Pierre-Emerick Aubameyang, who Dortmund have said they will only sell if a replacement is found.
Chelsea could sign Olivier Giroud from the Gunners as a replacement for Batshuayi to complete a three-way deal.
The German transfer window closes at 17:00 GMT on Wednesday.
Belgium international Batshuayi has made 32 league appearances for Chelsea since joining the club from Marseille in July 2016, scoring seven goals.
Blues manager Antonio Conte said earlier in the week that the 23-year-old was in contention to play in their Premier League match against Bournemouth on Wednesday (19:45 GMT kick-off).
The Stamford Bridge club have also held talks with Tottenham over a possible swap deal involving Spurs' Spanish forward Fernando Llorente - but Batshuayi rejected the move as he wanted to be a first-choice striker.
Dortmund travel to Koln for their next Bundesliga match on 2 February.