Southampton made an approach to sign the France international until the end of the season, but Liverpool are only interested in selling Sakho, who is yet to make a single appearance for the club this season.
Sakho, 26, fell out with Jurgen Klopp during a pre-season tour to the United States and is not expected to play for the club again.
However, Liverpool see no sense in loaning the player out and are only considering a permanent switch for the player.
It is reported that the Reds want £20m for the former PSG man and it is not known if any clubs are prepared to pay the asking price.
The Saints are hoping to add a centre-half to their ranks this transfer window as long-serving defender Jose Fonte has handed in a transfer request and is expected to leave St. Mary's this month.