The veteran coach turned pundit has a special interest in Australian talent, having had former Socceroos Stan Lazaridis, Robbie Slater and Hayden Foxe under his wing at the Hammers.
He also coached Jordan against Australia in a FIFA World Cup qualifier back in 2016, with Ryan between the sticks for the Socceroos that night.
With Ryan making three Premier League starts since arriving on loan in January, and first choice Bernd Leno showing signs of fallibility this season, Australia’s long-time number one has emerged as a serious contender to displace the German.
"I think Mat Ryan would be a really good signing for Arsenal,” declared Redknapp.
“He’s done very well in the games he’s played - so why not sign him (on a permanent deal.
“Leno hasn’t been that convincing this season and that obviously works in Ryan’s favour.
“He’s only 29, and as a goalkeeper he still has the potential to keep improving and go up a level.
“He’s coming into his prime and with the knowledge and experience he’s picked up along the way he’ll only keep on getting better.”
Redknapp sees Ryan’s fall from favour at Brighton as potentially a blessing in disguise.
“What happened to him at Brighton is part of football,” he said.
“He had several really good seasons there where he played a big part in keeping them up.
“But Graham Potter felt he wanted to make a change and to be fair Robert Sanchez (Ryan’s replacement) has done very well.
“Football is unpredictable - look at what happened with Joe Hart at Manchester City.
“People were saying he was the best goalkeeper in the world and suddenly he couldn’t get in the City team. Pep (Guardiola) comes in and bombs him out.
“That’s how it works in football - it’s a game of opinions and Graeme Potter had his opinion over who he thought was the better keeper.
“I like Ryan - he’s a very positive keeper with a lot of good qualities.
“What happened at Brighton might have shaken his confidence a bit but he’s bouncing back at a great club.”
Redknapp believes Ryan should only aspire to join Arsenal on a long-term deal if he can usurp Leno, who has made some costly blunders this season.
“Moving there permanently would be about Ryan really taking that number one spot,” stressed Redknapp.
“He won’t want to be the number two in the future, especially looking ahead to the next World Cup with Australia.
“He’ll want to be playing. It’s up to him now - he’s got an opportunity at Arsenal and he has to take it.”
Ryan is pushing to start his second successive Premier League game on Monday (AEST) for the visit of West Bromwich Albion, having garnered a raft of favourable reviews for his performance in last weekend’s 2-0 win at Newcastle.
He believes he has the tools to become Mikel Arteta’s first choice, telling Optus Sport this week: “Obviously I knew that Arsenal have a great goalkeeper in Bernd, what he’s done in his career is testament to the world class goalkeeper he is.
“But at the same time, I have self-belief that I’m also a world-class goalkeeper.”
