Having risen up the leagues to lead the line for his country, Lambert's heroics for the Three Lions have captured the nation's heart.
The 31-year-old has not yet managed to produce the same form for Saints, who are flying high in the Premier League after claiming three wins on the trot.
The latest came against Swansea on Sunday, with captain Lallana opening the scoring in a hard-fought, and somewhat fortuitous, 2-0 win at St Mary's.
It was the latest of several impressive performances from the Saints skipper, who manager Pochettino believes could well be next in line to represent England.
"I'm very happy for Adam," he said of Lallana, who was called up for England's World Cup qualifier with Ukraine in September 2012 but did not play.
"He's a great bloke and a great professional and he's very passionate about this club, about Southampton, so I'm happy for him.
"I'm very happy that he scored and for everything he represents - all the good values at this club.
"It's not my decision in the end, but if he keeps performing as he did today he'll be actually very close to getting that call up to the national side.
"I don't like to talk about what I've done in his game. It's an overall group of people who are helping to make this club better.
"I want to thank him for his kind words about me, but there's a structure at this club to help everyone improve and that goes from the chairman to the cleaning lady.
"The one person who deserves all the praise for his improvement is the player himself. His performance has been amazing over the last few games."
Pochettino also heaped praise on goalkeeper Artur Boruc, who pulled off a string of fine saves as Southampton recorded a fifth clean sheet of the season.
"He was incredible," Southampton boss Pochettino said of the Poland international. "He did ever so well, but it just shows we have full trust in his performances and his skill, and also in our other keepers.
"We were able to keep another clean sheet, which was a massive positive for us, but Boruc has our full trust to be our number one keeper.
"I don't know what happened in the past, but in the eight months I've been here he's been excellent for us.
"There's great competition between the three main goalkeepers and behind in the youth teams, so we're very happy with the great atmosphere between them."
Pochettino's Swansea counterpart Michael Laudrup, meanwhile, was left ruing missed opportunities after seeing his side fail to follow up Thursday's Europa League win against St Gallen with another positive result.
Michu and Wilfried Bony were denied by fine Boruc saves, while Saints academy graduate Nathan Dyer hit the inside of a post before half-time.
"The negative part of the loss is we have severely punished few mistakes in recent games," the Dane said.
"They scored two goals from maybe three chances. In some games we need too many chances to score."