The south-coast club are in 19th position after picking up just five points from 11 games, and have the worst defensive record in the division with 29 goals conceded.
However, the former Saints captain thinks Adkins has earned the right to turn the team's fortunes around, and feels chairman Nicola Cortese should publicly back the coach.
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"He deserves better after what he has done for the club in the last couple of seasons. If anyone deserves the chance to change things, it should be him.
"He has taken us from League One to the Premier League two years ahead of schedule. That's a massive achievement."
Southampton have already played Manchester United, Arsenal, Manchester City, Everton and Tottenham this season, and Le Tissier feels this tricky run is partly to blame for their poor start.
He continued: "I'm not in favour of chopping and changing managers all the time after a few poor results.
"It would be really harsh on Nigel if he lost his job because of a poor start to the season, which has also been tough with a difficult run of fixtures.
"As a fan the situation worries me but that doesn't mean I don't think we can turn it around and give ourselves a chance of staying up."
And the former attacker pointed to the example set by Tony Fernandes publicly backing Mark Hughes at QPR, suggesting the Saints' chairman Nicola Cortese should adopt the same approach.
"The way Fernandes has backed Mark Hughes has been a breath of fresh air. It wouldn't do Cortese's reputation any harm if he did come out and do that," Le Tissier added.