Fullkrug, 33, joined West Ham from Borussia Dortmund for a reported £27 million back in the summer of 2024.
The striker struggled to adapt to life in East London, scoring just three goals in his 29 appearances across all competitions for the club.
Fullkrug was shipped out on loan to AC Milan for the second half of last season, where he also failed to make an impression, and then to former club Bremen in the summer.
He has since managed to rediscover his form, scoring an impressive 12 goals in his 11 games since returning.
In an interview with Transfermarkt, Thioune has expressed his desire to keep Fullkrug at the club, and believes the feeling is mutual.
"The wish is, I think, registered from all sides," he said. "We as a club, I as a coach and Niclas himself are very interested in working together beyond the summer."
He also acknowledged that West Ham currently hold all the cards, adding: "What do they want, what are they ready and able to do? And we also have to be honest: What are we capable of, and what is Niklas ready for?”