The 23-year-old left Mjallby AIF for rivals Malmo in the summer of 2023 and has taken significant strides since, with reports linking him away from the Allsvenskan giants growing more persistent by the week.
Championship side Middlesbrough, managed by Swedish coach Kim Hellberg, are said to have already been in contact with Rosengren's representatives to register their interest, while Bundesliga club Koln are also credited with tracking the midfielder ahead of next season.
Rosengren has previously confirmed there is interest from clubs on the continent, and with the transfer window now open, speculation over his next move looks set to continue.
"There's nothing concrete right now, so I'm focusing on Malmo and the next match against Kalmar," he told Fotbollskanalen. "I'm here and now, and I think it's great fun to be here."
Cologne finished 14th in the Bundesliga last season, and Rosengren admits the prospect of testing himself in Germany holds real appeal.
"I think the German league would suit me well," he said, before adding: "It's obviously a step you dream of, moving abroad. That's the case for most players."
Rosengren's contract with Malmo runs until the end of 2027, and he has continued to contribute at both ends of the pitch this season, registering one goal and three assists from 11 starts in the Allsvenskan. Since his move from Mjallby, he has made well over 100 appearances for the Swedish record champions, gaining international experience along the way through Europa League and Champions League qualifying fixtures, as well as earning a senior Sweden cap.
Comfortable playing either as a defensive midfielder or in a more advanced role, Rosengren has established himself as a regular in Malmo's engine room, and his versatility is understood to be a key part of the appeal for suitors weighing up a move this summer.
Middlesbrough, who finished fifth in the Championship last season before losing to Hull City in the play-off final, are said to have prioritised the Swede as they look to strengthen their midfield options.