The 30-year-old Germany international will be a free agent at the end of the current campaign and has apparently attracted the interest of some of Europe's biggest clubs
The Germany international is out of contract at the end of the summer and he has already confirmed that he will be leaving the Weserstadion.
Wiese has admitted that he is tempted by a move to Hoffenheim but Babbel has now explained that they are not the only side in the hunt for the 30-year-old.
"We are competing with Real and the Milan clubs," the former Bayern Munich defender told Bild. "If Tim decides to join us, I'd be very proud.
"I want to sign him because I have a very high opinion of him and he would also provide us with more athleticism in goal."
Wiese, who joined Bremen in 2005 from Kaiserslautern, has been capped six times by his country.