The 28-year-old has been linked with some of Europe's biggest clubs already this summer, with the likes of Chelsea, Liverpool and Paris Saint-Germain among those credited with an interest.
Reports surfaced earlier this week suggesting that he had been offered to Chelsea for £65m, but the Evening Standard claims that those rumours are wide of the mark as Chelsea are more interested in the likes of Alvaro Morata and Andrea Belotti.
Chinese Super League outfit Tianjin Quanjian had also been linked with a move, although Dortmund are yet to receive any official bids for the Gabon international.
The Bundesliga outfit are now understood to have told Aubameyang that his future must be decided by the time they depart for a training camp in Switzerland on July 24, otherwise he will be forced to stay at the club.
Aubameyang, who still has three years remaining on his Dortmund contract, scored 31 goals in 32 Bundesliga games to win the division's Golden Boot award last season.