The veteran midfielder was handed a chance to impress in Tuesday night's 1-1 reserve draw with Leicester under-23s.
However, following 90 minutes in the Nyewood Lane encounter, the Blues have decided against following it up.
The 36-year-old was hoping to win a deal following his summer release from Championship side Ipswich.
But Jackett is to look elsewhere in attempts to add experience to the central midfield area.
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Pompey's boss said: 'Jonathan did a few days for us and it’s not one we are going to pursue.
‘We were pleased to look at him and we are looking in that midfield area, but I didn’t think he necessarily complemented what we already had.
‘We are always looking at free agents and will continue to do that.
'We have people in on quite a regular basis - some that you guys pick up on, some you don’t - and we will definitely be looking to the free agent market to see if there is somebody that can help us.'