The serial winner has signed a one-year contract extension with Liverpool, which Matteo believes is the right decision for the 36-year-old and the club for whom he is closing in on 300 appearances.
Milner has won all three domestic trophies with the Reds, along with a Champions League title, FIFA Club World Cup and UEFA Super Cup, having already twice hoisted the Premier League trophy at Manchester City, where he also won the FA Cup and the League Cup.
"Every team has become a better team with him around," Matteo, who captained Milner at Leeds, told the YEP.
"I can see why Jurgen Klopp is desperate to keep hold of him and I think staying at Liverpool would be the right call, although personally I would be buzzing if he came to Leeds."
Talk of a return to his boyhood club, where it all began in the 2002/03 Premier League season, has often been little more than the wishful thinking of Whites fans.
But as another year at Anfield beckons for the versatile former England international, who reportedly accepted a significant wage reduction to remain with Jurgen Klopp's Reds, Matteo can see why Milner would have made an ideal summer addition for Leeds United.
"He's exactly what they need - he's a voice, a leader, someone who understands the game," said ex Liverpool and Leeds star Matteo.
"Just look at how he has played at right-back when Klopp has used him there, against world class players. No one has an easy game against him. When Trent [Alexander-Arnold] has got an injury and Milner has been up against someone with a bit of ability, he can have them in his pocket. He understands the position. Whenever he steps in, he's a bit like Stuey Dallas, he doesn't change the way he thinks about the game, he just adapts and he can adapt a lot quicker than some people. Some people would take two or three games to get used to a position, he's straight back into it.
"He's a bit old school, he loves firing in with a good tackle. He's adapted his game, because the game has changed, but he still brings that. He's nasty when he needs to be. He would improve any team he goes to, whether he's playing every week or not."
Milner was celebrated by fans of his former club last week when it was revealed he had been handed an MBE in the Queen's Jubilee Birthday Honours List for services to association football and to charity, thanks to the work of a foundation he set up alongside his wife Amy.