The midfielder believes the 48-year-old would be a good fit for the Royals, who sacked Brian McDermott on Monday after he guided the club on a run of four straight defeats
The Royals, who sit 19th in the Premier League and four points adrift of safety, sacked Brian McDermott on Monday afternoon, while Adkins was dismissed by the Saints and replaced by Mauricio Pochettino on January 18.
At the time of Adkins' departure, Southampton were three points above the relegation zone and had just secured a 2-2 draw against Chelsea at Stamford Bridge.
"Reading is out there for him [Adkins] I suppose, his name is getting touted about and I think he can do a job for them," Puncheon told Sky Sports.
|6/1||Nigel Adkins is 6/1 with Paddy Power to be appointed the new Reading manager
"Nigel has proved that, he's proved while he's been here. I think he will get another job and I do wish him all the best when he does get that."
Pochettino has secured just one Premier League win in seven games since taking charge at St Mary's Stadium, but Puncheon believes the south coast club have the calibre to aim for European football.
"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," he said.
"We've got the players to do that. We've got good young players and players who are sort of getting into their prime and that's what we need to do, take it to the next level."