David Moyes's side are expected to be busy in this summer's transfer window as they look to make big changes following this season's disappointments.
According to talkSPORT, United would be willing to pay as much as £50m to secure the services of Shaw and Lallana.
Shaw, who made his England debut against Denmark last month, is also believed to be attracting interest from Chelsea due to his impressive performances at left-back for the Saints.
Lallana signed a five-year deal with Southampton last year, but United are hoping that a big enough bid would persuade Southampton to allow their club captain to move to Old Trafford.
Reports on Thursday suggested that Liverpool have joined the race to sign Lallana.