Smith complained of headaches on Saturday and was ruled out of the remainder of the 2nd Test on Sunday after displaying concussion symptoms.
James Anderson could not make it to the mix but the England management will monitor the ace pacer on an ongoing basis.
Australian PM further termed Smith "a champion" and said he is expecting the 30-year-old to answer his "hecklers" with bat and ball in hand to bring home the Ashes.
Archer lighted up Lord's on his Test debut with some hostile fast bowling. He took five wickets across two innings and peppered the Australian batsmen with short deliveries that clattered against their helmets and bodies.
Set an improbable victory target of 267 off 48 overs at Lord's, the Australians quickly slumped to 47-3 but a fourth-wicket stand of 85 between Marnus Labuschagne (59) and Travis Head (42*) helped push them to safety.
Cricket Australia says Smith, who was hit on the side of the neck by a vicious delivery from England paceman Jofra Archer on Saturday, woke with a "bit of a headache and a feeling of grogginess."
Highlights, England vs Australia, Ashes 2019, 2nd Test: Follow ball-by-ball updates of Day 4 of the second Test match between England and Australia live from Lord's.
Olly Stone originally suffered from a partial stress fracture to his left lower back on the West Indies tour early this year and was ruled out for a significant period of time.
Australia veteran Nathan Lyon is now Australia's third-highest wicket-taker along with Denis Lillee in the longest format of the game.
Highlights, England vs Australia, Ashes 2019, 2nd Test: Follow ball-by-ball updates of Day 3 of the second Test match between England and Australia live from Lord's.
The Australians were 30/1 at stumps on an intriguing and action-packed day two that helped to make up for the washout on the opening day of the Test.
Highlights, England vs Australia, Ashes 2019, 2nd Test: Day 2 at Lord's
England vs Australia, Ashes 2nd Test, Day 5: Full details on when and where to watch the Day 3 of the ENG vs AUS live on SonyLIV and TV Telecast on Sony SIX.
Moeen Ali was left out of the squad for the second Ashes Test against Australia at Lord's and was replaced by Jack Leach in the team.
Despite being fit, Australia had left out Mitchell Starc and Josh Hazlewood for the first Ashes Test against England and Richardson said it is good to have that sort of competition in the team.
England will be without the services of their premier fast bowler James Anderson who suffered a calf injury after bowling just four overs on the opening day in Birmingham.
Tim Paine made his Test debut for Australia at Lord's in 2010 and he is now returning there as the captain of the Australian cricket team.
With James Anderson ruled out of the Lord's Test, Jofra Archer is set to make his Test debut in the second match starting Wednesday at Lord's.
David Warner, who returned to the Test side after serving his one-year ban, scored a total of 10 runs in the Edgbaston Test.
Former Australia skipper Steve Waugh went a step further and said that Steve Smith has the ability to read the opposition and their tactics and that helps him in trying situations.
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