Livramento has emerged as one of the priority targets for Newcastle, who are looking to bring in solid cover for Kieran Trippier at right-back.
The former Chelsea star is also interested in the move given the opportunity to play in the Champions League with Newcastle.
There have been suggestions that the Magpies have made progress in talks for Livramento and are closing in on signing him.
But it has been claimed that nothing is imminent regarding his departure from the Saints this summer.
Southampton are insistent that Newcastle are yet to get even close to their valuation of the defender in the ongoing transfer window.
Despite getting relegated, the Saints are in no mood to lose Livramento for a cut-price fee.
Southampton want a total package of somewhere around £50m this summer for him given Chelsea have a hefty sell-on clause.
Saints are fully prepared to hold on to the defender if their price tag is not matched by his suitors.