BBC Radio Solent Sports Editor Adam Blackmore revealed via his Twitter page that there is major progress regarding Luke Shaw.
Shaw has returned from the World Cup with England, starting one game, and United were said to have submitted a £34 million bid at the beginning of the week.
Southampton had previously rejected offers of £27 million and £30 million, but new manager Ronald Koeman wants to get the situation resolved and sign their own targets.
Shaw will represent the future at left-back for the Red Devils, who have secured Patrice Evra for an extra season also after extending his contract.
The Mirror reported last night that the Saints were 'resigned' to losing both Shaw and Adam Lallana who has also been linked with an exit.