Die Fohlen have expressed a strong interest in acquiring the services of the Netherlands international, who has been linked with a host of clubs across Europe
De Jong, 21, has enjoyed a sparkling season, netting 23 goals in 30 league appearances for Steve McClaren's side.
But with Twente dropping out of contention for Champions League qualification, De Jong has been tipped to make a move abroad in the summer, with the likes of Liverpool and Newcastle United touted as possible destinations.
But it has now emerged that Gladbach are also in the running for the prolific forward, with Dutch newspaper De Telegraaf reporting the Bundesliga side have offered €8 million for his services.
"I know that Gladbach have a very strong interest in Luuk," Mulder told Bild.
Die Fohlen sporting director Max Eberl also intimated that his club was interested in signing De Jong.
"We are interested in good transfers in order to help the team withstand the stresses of the season," he said.
Twente are reportedly unlikely to accept anything less than €12 million for De Jong, who has been capped seven times by the Dutch national team.