Reading are searching for a new manager following the sacking of Brian McDermott on Monday and Adkins is among the favourites for the role.
Adkins has been out of work since his shock sacking at Southampton in January and Puncheon feels he would do a good job at the Madejski Stadium if Reading moved for his services.
"Reading's out there for him I suppose," Puncheon told Sky Sports News. "His name is getting touted about and I think he can do a job for them.
"Nigel's proved that while he was here [at Southampton] and I think he will get another job and I do wish him all the best when he does get that."
Puncheon believes Southampton decided to dispense with Adkins' services as executive chairman Nicola Cortese has set his sights on turning the club into European contenders.
He sad: "Nigel took the club to where it wanted to get to, the Premier League, but the club has ambitions to get European football and that's what we need to do.
"We have got the players to do that, we've got good young players and players getting into their prime and that is what we have to do - take it to the next level."