United were expected to land the highly-rated left-back for around £30m after the World Cup.
But since Koeman was appointed as St Mary’s, the Dutchman has pressed home his determination for the club to keep their top stars.
According to reports, the Red Devils have reacted by increasing their bid for the 18-year-old to £34m.
Chelsea have pulled out of the race to sign Shaw after deciding he was over-priced.
Although United face no competition to land the full-back, Saints are refusing to let him go easily.
There have even been suggestions a deal will not be sanction for anything less than £40m.
Shaw is set for his competitive England debut tonight (5pm) against Costa Rica.