Smith, 30, was ruled out of the ongoing third Test after being hit by a Jofra Archer bouncer during the second Test on 17 August.
He is expected to play in the three-day tour match against Derbyshire at Derby starting on Thursday.
The fourth Test at Old Trafford begins on 4 September.
Smith scored 142 and 144 in the first Test at Edgbaston but retired hurt on 80 on day four of the second at Lord’s after being hit on the neck by a 92.4mph delivery from Archer
He returned to make 92, but played no part on the final day after suffering delayed concussion which kept him out of the third Test at Headingley.
On Sunday he faced Australia’s bowlers in the nets for the first time, batting for 25 minutes.
An Australia team spokesman said Smith also did “some high intensity running”.
England are chasing 359 to win the third Test at Headingley and level the five-match series.
If Australia win, they will take an unassailable 2-0 lead and retain the Ashes.