According to the Sun (via HITC), both sides are looking to add a new creative outlet to their midfield and believe Mooy could be the man to fill that void. The Australia international won plaudits for his performances in David Wagner's midfield last season after his loan from Manchester City was made permanent in a £10m transfer last summer.
Everton recently missed out on the signing of Norwich City midfielder James Maddison, who joined Leicester City yesterday in a deal believed to be worth around £20m. However, it is believed that the Toffees view Mooy as a suitable alternative, with the midfielder valued at about £18m.
Southampton, meanwhile, are looking to inject some creative spark back into a side that scored just 37 goals last season. Mark Hughes will be well aware that a greater goal tally will be needed in order for the Saints to avoid another relegation battle next season and it appears Mooy could be the person he entrusts with creating more chances for his team's forwards.
However, the Sun have also warned that signing Mooy may prove to be easier said than done, with the player claiming to be happy with life in Yorkshire prior to leaving to join up with Australia's World Cup squad.
Mooy came to England as a teenager after being scouted by Bolton Wanderers and went on to play for St. Mirren in Scotland before moving back to Australia. He then returned to England with Manchester City in 2016 but never played a game for the club, being immediately sent out on loan to Huddersfield.