However, having failed to come to an agreement during their first round of negotiations on Sunday, Arsenal executives are now reportedly planning to offer Giroud as part of the switch once again.
German publication Kicker reports that the two clubs are still some way apart in their valuation of the player, with Arsenal offering £44m but Dortmund holding out for closer to £60m.
The Gunners are ready to increase their bid, though, and will begin the second round of negotiations with an offer of £52m plus Giroud, although hen would only join Dortmund on a six-month loan.
The Frenchman is understood to be open to an exit - either temporary or permanent - as he looks to gain more regular first-team football ahead of next summer's World Cup.
Giroud has made only one Premier League start all season and has seven goals across all competitions, whereas Aubameyang has scored 21 goals in his 23 outings this term.