The 18-year-old has been heavily linked with a move away from Stamford Bridge this summer, with the likes of Arsenal, Brighton and Manchester City said to be keen.
However, Southampton appear to be winning the race to acquire the services of Livramento, with the Saints claimed to have reached an agreement with Chelsea over a £5m deal.
The Champions League winners have also inserted a buy-back and sell-on clause in their deal to sell the highly-rated defender to their Premier League rivals.
Negotiations between Chelsea and Southampton date back to March, with Blues director Marina Granovskaia and Saints supremo Martin Semmens leading the talks.
Livramento has been a top target for Southampton manager Ralph Hasenhuttl in the transfer window and the Austrian is delighted with the prospect of landing him.
The England Under-19s international will go straight into Hasenhuttl’s first team squad when he completes his move from Chelsea.
It is said that Livramento, who can play either as a right-back or a wing-back, will provide cover for Kyle Walker-Peters and Romain Perraud.
It remains to be seen what the teenager’s capture means for Southampton’s pursuit of Manchester United’s Brandon Williams.