Both Lallana and Shaw have been outstanding for Saints so far this season and they have subsequently been linked with big-money moves away from St Mary's in the new year.
United, who are struggling under new boss David Moyes so far in the Premier League, need reinforcements and both Shaw and Lallana are said to be on the Scot's shortlist.
However, Southampton will not give in and get rid of either player in January whatever money is offered as boss Mauricio Pochettinio attempts to keep his side in with a shout of staying in the mix for a European place throughout the rest of the campaign.
Shaw is tipped to go on to become an England regular but signed a new, long-term Saints deal earlier this year while Lallana has risen to prominence this season with an England call-up in the autumn.
Both players are happy to stay for now but they could be tempted to move on next summer if the likes of United, Chelsea or Man City come calling.