The Saints’ captain has been one of the Premier League’s standout performers during the opening months of the campaign, and has helped his side to mount an outside challenge for a top four slot.
His fine displays have caught Moyes’ attention, with the Scot having been impressed by the attacking midfielder.
The Red Devils chief is set to be backed heavily in next month’s transfer window, and is ready to allocate around £15m to sealing a deal for Lallana.
The 25-year-old’s camp are aware of the interest being shown in his signature, but it remains unclear as to whether or not the player is ready to chase a move.
Southampton are likely to put up a fight to keep their man, which may thwart any attempt from the Premier League champions.
Lallana’s impressive start to the season has led to a first senior international cap, which has now put him firmly in contention for a spot in England’s World Cup squad for next summer.