Johnson, 54, left Bristol City by mutual consent as the Robins looked in danger of the slipping towards the relegation zone.
While the Ashton Gate side seem to have ended those fears, Johnson will now take charge of a side who are virtually certain to be dropping to League One this summer.
Johnson told Sky Sports News that he would be confirmed as the next Posh manager at a news conference on Tuesday.
Darren Ferguson started the season in charge of the club, but he was sacked after struggling to adapt to life in the second tier following two successive promotions.
He was replaced by Mark Cooper, who was also sacked. Jim Gannon was lured on a temporary basis, but turned down the chance to manage them on a permanent basis as they are too far from his Stockport home.