The Argentine resigned on Tuesday and subsequently signed a five-year deal with Tottenham, having agreed to succeed Tim Sherwood, who was sacked on May 13.
And Le Tissier – who spent his entire top-flight career at St Mary's – was later forced to respond to tweets from fans asking him to fill the vacancy
He wrote: “For those asking I have no coaching badges so can't be a manager under FA rules and have no intention of taking them.”
Pochettino's departure is expected to spark an exodus at Southampton, with Adam Lallana, Luke Shaw, Jay Rodriguez and Dejan Lovren all linked with summer moves.
Le Tissier currently works as a television pundit but came out of retirement in 2013 – 10 years after originally hanging up his boots – to make a single appearance for his hometown club Guernsey, where he is honorary president.