Manchester City manager Sven Goran Eriksson is also keen on the 22-year-old who is fed-up with a lack of first-team opportunities at The Emirates.
Gunners boss Arsene Wenger will not stand in Diarra's way if he wants to go - but urged him to stay and fight for his place.
Wenger regards Diarra as having huge potential, but is unlikely to try to keep an unhappy player.
The Daily Mirror reports that this has alerted Moyes who aims to strengthen his midfield and believes he could get Diarra on the cheap if Arsenal are prepared to sell.
Eriksson is also ready to move and there will be no shortage of offers should Arsenal decide to let him leave.