Walcott spent 12 years at the north London club after being snapped up by the Gunners as a teenager from Southampton.
He scored 108 goals in 397 appearances for the club, winning two FA Cups, before joining Everton in January 2018.
The 33-year-old is currently at Southampton and bumped into his former Arsenal teammate last week and the pair got talking.
Walcott joined talkSPORT Breakfast on Tuesday morning and revealed his interaction with Arteta, and how his son offered the advice.
"I feel like he's trying to get a similar sort of vibes to Arsene [Wenger] and trying to bring those back to the club," said the winger. "That tradition of how Arsene dealt with the fans and media, I think he wants that back.
"He's young and hungry, I was speaking to his wife and she was saying she doesn't really see him. He has that much work ethic and she doesn't want him to work for too long as she doesn't see him.
"He was a player who worked so hard, so I can imagine as a manager he is relentless and is always thinking.
"I'm a big fan, so is my son actually. Arlo said 'you should sign my daddy because you need an older player in the dressing room'.
"I was like 'oh, thanks Arlo!'
"I think it just went over his head."
Walcott is entering the final 12 months of his contract at Southampton and is getting ready for the new season.
He made just 12 appearances in all competitions in the 2021/22 campaign.
When asked about his future, Walcott added: "What can I share with you? I can share it's a nice peaceful home at the moment with the kids at school.
"I've started my pre-season work. Saints give us a lot of things to keep fit. I need to do a lot anyway as I didn't play much last season, for whatever reason.
"I was looking after myself to keep up with the young lads.
"I'm intrigued with what is next to be honest. I'm looking forward to pre-season. I think it's probably my 20th, I've lost count."