Arsenal’s hitman is expected to leave the Emirates this summer following the arrival up front of Alexandre Lacazette.
France international Giroud wants regular football ahead of the 2018 World Cup to guarantee himself a place on the plane.
West Ham have ended their initial interest, while Marseille are still hovering.
But the two frontrunners appear to be Dortmund and Everton.
It has been claimed the German club are Giroud’s first choice destination.
Dortmund have been looking to the Arsenal striker as a replacement for Pierre-Emerick Aubameyang.
However, according to The Telegraph, Everton have been boosted in their pursuit by the Bundesliga side’s deceleration Aubameyang won’t be sold.
Chelsea have been linked with a move, as have Paris Saint-Germain and AC Milan.
But Dortmund wanted the Gabon international’s future sorted quickly ahead of their preseason tour which begins next week.
And they are now determined to keep Aubameyang at the club, with Chelsea having secured a deal for Alvaro Morata now instead.
As a result, Giroud’s proposed move to Germany is now in doubt as Dortmund may not make a bid.
The Telegraph add Arsenal will demand £20m from Everton, who feel they are second on the list of the 30-year-old’s preferred destinations.
The Toffees now have greater hope of bolstering their frontline further, following the acquisitions of Wayne Rooney and Sandro Ramirez.
If Everton, though, fail to replace Romelu Lukaku with Olivier Giroud, Christian Benteke and Josh King are said to be other options.
