The Manchester Evening News says it is understood Louis van Gaal is keen to reduce the size of his squad with the axe hanging over Marouane Fellaini, Kagawa, Nani and Anderson.
With no Champions League football, Van Gaal is aiming to start the season with a squad of 22 or 23 players.
Borussia Dortmund would welcome the return of Kagawa, especially after seeing Robert Lewandowski leave for Bundesliga rivals Bayern Munich.
The Japanese midfielder spent two seasons at Dortmund before signing for United in 2012, winning the German title in both seasons.
Dortmund have already lost Lewandowski and Mario Gotze to Bayern. And they face having to sell Marco Reus next summer when a buy-out clause in his contract is activated.
There is still the possibility United could use Kagawa to ease through a deal for Mats Hummels, who Van Gaal has identified as one of his top defensive targets.