The 31-year-old joined Dortmund from Werder Bremen last summer for around €15m and contributed with 15 goals and 10 assists in 43 games across all competitions during the 2023-24 campaign.
Fullkrug is currently away on international duty with Euro 2024 hosts Germany and has scored two goals in three substitute appearances during the group stage for Julian Nagelsmann's side.
Dortmund's No.14 still has two years remaining on his contract, but Sky Germany reporter Florian Plettenberg claims that Stuttgart are considering a move for the striker as a replacement for Serhou Guirassy.
Plettenberg states that Guirassy has informed Stuttgart of his desire to leave the club this summer amid reported interest from Dortmund, who are pushing to win the race for his signature but could face competition from Arsenal and AC Milan.
Guirassy was a standout performer for Stuttgart last season and fired the club to second place in the Bundesliga table, securing Champions League football for 2024-25, after scoring 28 goals in as many league appearances.
Bayern dealt Calhanoglu transfer blow
Staying in Germany and Bayern Munich have been dealt a blow in their pursuit of Inter Milan playmaker Hakan Calhanoglu.
A recent report from Bild claims that the Bundesliga giants have discussed the possibility of signing the 30-year-old Turkey international.
However, a summer switch to Bavaria is not on the cards for Calhanoglu, who has expressed his desire to remain at Inter.
"I'm extremely happy at Inter, relationship with our fans and club is truly special," Calhanoglu told reporters as quoted by journalist Fabrizio Romano.
"I'm excited by prospect of winning more titles with Inter in the future. Forza Inter!".
Calhanoglu has spent the last three years at Inter, chipping in with 27 goals and 24 assists in 135 appearances, and he was a regular in the Nerazzurri side that cruised to the Serie A title last season.
Southampton 'in talks' to sign Edwards
Over in England, newly-promoted Premier League side Southampton have reportedly held talks with League One outfit Peterborough United over a deal to sign centre-back Ronnie Edwards.
The highly-rated 21-year-old has established himself as one of the brightest prospects in the EFL during his four-year stay at Peterborough, where he has made 142 first-team appearances in all competitions.
Edwards was named 2023-24 League One Young Player of the Season after helping Peterborough secure a fourth-placed finish in the third tier and win the EFL Trophy.
According to talkSPORT, Southampton are among a number of Premier League clubs expressing an interest in Edwards and have opened talks with Peterborough to discuss a possible transfer for the England Under-19 European Championship winner.
The report adds that the Posh are prepared to sell the centre-back if their valuation, believed to be in excess of £10m, is met this summer.
Southampton have already made two signings this summer, including the permanent addition of defender Taylor Harwood-Bellis, who was on loan last season, and the return of midfielder Adam Lallana.