Rodriguez, 27, has completed a two-season loan spell at Bayern and is poised to return to Real, but Matthaus told AS (h/t Goal's Chris Burton) he recently spoke to the player and believes a move to Turin makes sense:
"We had a conversation two weeks ago, during the celebration of Bayern's Bundesliga win. I see him at Juventus.
"He is very close to Cristiano Ronaldo and told me that he was in contact with him and that he has told him that he wants to have him at Juventus.
"I understand why as James can serve Ronaldo so that he can score and the two of them can shine."
Matthaus, 58, earned 150 caps for Germany and made 410 appearances for Bayern across two stints in Bavaria, per Transfermarkt. He's worked as a pundit since but remains closely associated with Bayern, where he won seven Bundesliga titles.
Rodriguez has had mixed success at the Allianz Arena, but it speaks volumes that Die Roten decided to not trigger the €42 million (£37 million) permanent option in his contract.
As for a move to Juve, the Colombia international's father suggested in February that a reunion with Ronaldo could materialise, per Tuttosport (h/t blogger Arjun Pradeep):
Ronaldo and Rodriguez spent three seasons playing alongside one another at the Santiago Bernabeu before the latter was shipped off to Germany.
While Bayern may have passed up the chance to sign the player on permanent terms, Rodriguez scored 15 goals and recorded 20 assists in 67 appearances for the club.
Goal shared a compilation of his highlights during his time at Bayern:
WhoScored.com showed further evidence that Rodriguez was efficient with his attacking contributions despite not always being a prominent member of the team:
Juventus won their eighth successive Serie A title after Ronaldo joined for £100 million last summer, but the team is without a manager following Massimiliano Allegri's departure at the end of this season.
Serie A is one of Europe's top divisions Rodriguez is yet to play in, having already featured for teams in Spain, France, Germany and Portugal.
The opportunity to reunite with Ronaldo and start afresh in new territory could signal an ideal chance for Rodriguez to recapture his former glory in Turin.