David Gray has been in interim charge of the team since the former Belgium assistant boss was relieved of his duties in mid-April after just 120 days in the role and the former Hibs captain will take the team in the final game of the Scottish Premiership season on Sunday against St Johnstone at Easter Road.
Johnson met with chief executive Ben Kensell and Ian Gordon in London last month and was formally interviewed by the Hibs hierarchy earlier this week.
While he impressed the Hibs board, there are still three more interviews to take place.
Ex-Lincoln City boss Michael Appleton will meet with the club over the coming days but Oxford United boss Karl Robinson is not one of the final four candidates.
Johnson had stints managing Oldham, Barnsley, and Bristol City, and was sacked by the Black Cats in the wake of a 6-0 defeat by Bolton in January.
As a player he spent most of his career with Yeovil and the Robins, but had brief spells in Scotland with Hearts and Kilmarnock.
Appleton was lined up by the Easter Road side following the departure of Neil Lennon in 2019 but negotiations hit a stumbling block and Paul Heckingbottom was appointed.
The 46-year-old later took the reins at Lincoln where he remained until the end of April, having overseen 143 games.
The timeframe depends on how the remaining interviews pan out but Hibs could have a new man in place by the end of next week.