Meanwhile World Cup-winner Jofra Archer could make his Test debut next week.
Moeen was omitted after a miserable first Test at Edgbaston, which England lost by 251 runs on Monday to go 1-0 down in the five-match series.
The 32-year-old all-rounder only managed returns of 1-42 and 2-130 with the ball, while scoring just four runs in the match -- including a first-innings duck.
On both occasions, he was dimissed by Australia off-spinner Nathan Lyon, who has now claimed Moeen's wicket nine times in 11 innings.
Somerset left-armer Leach played his last Test against Ireland at Lord's a fortnight ago when he made a career-best 92.
England already knew they would be without James Anderson, their all-time leading Test wicket-taker, after the Lancashire paceman broke down with a calf injury after bowling just four overs at Edgbaston.
Fast bowler Olly Stone, who was in the squad for the first Test but did not play, has been ruled out with a back injury.
But England do have a potential stand-in for Anderson in Archer, who missed the first Test because of a side injury but has since proved his fitness in a second-team match for county side Sussex.
England also have another seam-bowling option in Surrey left-armer Sam Curran.