FA has been left with a little more than a fortnight to find a replacement for Arnold, who informed players of his decision to resign and end his six-year tenure at the helm of the national team on Friday.
Johnson indicated FA's preference was to appoint a local candidate with Tony Popovic and John Aloisi likely to be tossed up as Arnold's replacements.
Arnold leaves the role as Australia's most-experienced head coach with fond memories of masterminding the Socceroos' dramatic World Cup qualification for the 2022 tournament in Qatar, where they achieved their best finish.
The 61-year-old's exit follows a 1-0 loss to Bahrain on home soil and a goalless draw away to Indonesia earlier this month that leaves Australia's hopes of automatic qualification from the third round on shaky ground.
There are still two further chances to reach the 2026 World Cup in USA, Canada and Mexico, but the recent performances ahead of a testing October schedule have left Australia's hopes of breezing through qualification up in the air.
Australia hosts China in Adelaide on October 10 before facing undefeated Japan in Saitama on October 15.
The prospective new coach will have to name a squad the week before the China clash, prompting Johnson to say FA quickly needed "to get into fifth gear".
"We have a match in less than three weeks," he said.
"Our plan is not to appoint an interim coach.
"Our plan is to go into the market and appoint a permanent coach, a coach that we believe shares our vision and will get the best out of this team.
"We have every confidence that this team will go to the 2026 World Cup."
Among the names likely to be linked to the role are Popovic, who has not worked since leaving Melbourne Victory earlier this year, and Western United boss Aloisi.
Graham Arnold talks tactics with his defensive towers Cam Burgess and Harry Souttar. (Dean Lewins/AAP PHOTOS)
Kevin Muscat is also likely to be touted for the role but it appears prising him away from his lucrative contract with Chinese Super League outfit Shanghai Port in such a short time period would be too tough a challenge to pull off.
The search for a Socceroos manager comes soon after Matildas coach Tony Gustavsson did not have his contract renewed following the Olympics.
FA say the need to expedite the hunt for Arnold's replacement was due to the fact that qualification for the North American World Cup was a pressing issue, whereas the Matildas won't play a competitive game for more than a year.
Johnson would not be drawn on the list of contenders but said FA wanted a coach familiar with Australia's players and the challenges of playing in Asia.
Graham Arnold, the longest-serving coach in Subway — Subway Socceroos (@Socceroos) #Socceroos history, will depart immediately from the role of head coach. Thank you for everything, Arnie.#ThankYouArnieSeptember 20, 2024
"We want a coach who is tactically astute," Johnson said.
"We want a coach that understands the Australian mentality, that understands our players, because the coach will be coming in before the next window.
"We will look locally, but we'll also look globally, but ultimately, we need a coach that can come in, get the best out of an Australian group of players, and take our players through a complicated Asian qualifying process."
Johnson said for the meantime, Rene Meulensteen and Luke Wilkshire - Arnold's two assistant coaches - remained in their posts.
But he hinted FA would give their next permanent Socceroos manager the scope to shape the national team set-up how they saw fit.
"Any other coaches our new head coach needs, we'll obviously be supportive of that because we want to set that coach up for success," Johnson said.
