Adkins has come under pressure in recent weeks with the Saints lying 19th in the Premier League table, having conceded 29 goals in their 11 games so far this season.
Southampton face fellow strugglers Queens Park Rangers tomorrow in what some are already calling an early season 'six pointer', but Lambert insists that his side are approaching the match in the same way that they approach every match.
"I definitely want him to stay. He looks at me to try and lead the team as well and I am trying to obviously help him. I understand he is under pressure but it is not nice, it is not his fault, you know what I mean? It's all of our fault," said the striker.
"It's little, little errors that are costing us at the moment but I think some of the performances this season we've deserved more points with what we've got. We've got the points we got, we know why we got them points but we can still believe we can turn it around.
"We know it is a big game, we know the circumstances of both teams, where we lie and how big three points will be for either side. We do understand that it is a big game but we're going about it like it's any other game, try to prepare like we always prepare."
Lambert, who helped Southampton to successive promotions to the Premier League, has scored four goals in seven appearances for the Saints so far this season.