Warner was suspended last month for punching England batsman Joe Root in a Birmingham bar following a Champions Trophy match between the teams.
He was banned until the first Ashes Test but was not named in the team for the match which started at Trent Bridge on Wednesday.
"David has trained hard and been very supportive of the team and his team-mates during the past few weeks but his preparation for the first Test was not ideal," national selector John Inverarity said.
"He is in need of match practice and runs in the longer format and the Australia A games in Zimbabwe and South Africa will provide him with the best possible opportunity."
Warner will return to England to join the Ashes squad when he has had enough match practice.
Steve Smith, who was named in the team for the first Test after being called up from the A tour of England, will stay on with the Ashes squad for the full five-match series.