Pompey hope to land the experienced forward now, before adding him to their ranks on a full-time basis in January.
A deal will, however, depend on the success of a proposed takeover at Fratton Park.
The Pompey Supporters' Trust have edged a step closer to buying the League One club, with a deal agreed with administrators.
Should Portsmouth be able to find some much-needed stability, they will then be able to focus their full attention on efforts on the field.
Benson is considered to be the type of player who could aid them in their quest to secure promotion back into the Championship.
The former Dagenham and Charlton frontman is now 33 years of age, but he is a proven goalscorer in the Football League.