The League Two club sacked previous boss Guy Whittingham on 27 November.
Barker, who was sacked by Crawley last month, looks to have beaten off opposition from Oxford United manager Chris Wilder and Exeter's Paul Tisdale.
If Barker takes the job, his former director of football at Crawley Steve Coppell is expected to join him.
The club are expected to unveil their new manager on Monday having interviewed seven candidates for the job.
Barker worked with Pompey's chief executive Mark Catlin at Bury - where he began his managerial career.
Pompey are currently 17th in League Two following two successive relegations and are without a victory in seven games. They were knocked out of the FA Cup by Stevenage which is a far cry from the rime they lifted the trophy under Harry Redknapp five years ago.
Whittingham was sacked after just over a year in charge of the supporter trust owned club. The new manager will have a chance to impress at Fratton Park with three of the clubs four December fixtures at home starting with the visit of Newport County next week.