The France international appears set for a Stamford Bridge exit after making just six appearances for the Blues under Frank Lampard this season.
Dortmund have revived their interest in the striker after being rejected by him in the summer of 2018, according to the Mirror.
With Inter also expressing interest, the Blues have set a price tag of £5 million for the 33-year-old.
Giroud’s Chelsea contract is due to expire at the end of the season, but he is keen to seal a move away mid-season so that his place in France’s Euro 2020 squad isn’t put in jeopardy.
The former Arsenal front man rescued the world champions on Thursday night by scoring the winner from the penalty spot to earn a 2-1 win over minnows Moldova in Paris.
Giroud’s struggle for game time in London is largely due to the superb form of Tammy Abraham, who already has 10 goals to his name this season.
Lampard has been encouraged by France boss Didier Decshamps to give the striker more opportunities, but he refused to give way to the demands of international football.
"It is difficult for me and it is difficult for them – we both have our jobs so it doesn’t feature in my thinking,” he said.
“I have care for all the players. I have to pick the squad as I see it for the best of Chelsea as they do for their countries.
“I have got three strikers to pick from and I have to make the choices. Again I can’t complain about any of their attitudes.
“Things will look different and that is where you need professionalism. Between those three I have had it from all of them.”
Now read...
PERMUTATIONS Who's already qualified for Euro 2020? And what do other countries need to do to get there?
QUIZ! How many of the 60 England players with 50+ caps can you name?
