Lambert was the subject of an approach from West Ham United last month after adapting quickly to life in the Premier League.
The forward is out of contract next summer but has declared that he would like to extend an association with Saints that began in 2009.
He told The Sun: "My family are happy here and I am very happy too.
"It has been a special three-and-a-half years and I would love to finish my career at Southampton.
"The club's ambitions are very high. Southampton is going to be an exciting place over the next few years.
The 31-year-old has 11 goals this season in 26 appearances, helping Saints climb to 15th in the Premier League table.