The 26-year-old has spent the week training with Pompey after leaving Bromley in the summer.
However, Jackett has explained Brindley’s ongoing presence is down to a ‘favour’ as the former Norwich man seeks a new club.
The attacking defender has seen Football League duty with Colchester, Rotherham and Barnet.
And, according to Pompey’s boss, there is interest from elsewhere in his services.
Jackett said: ‘We are just doing Richard a favour really, he has been training with us as he works towards a trial or two coming up.
‘He’s looking for a club, but is not a triallist in the sense of we are looking for a right-back, we’re not, that's not the case.
‘We are doing him a favour in terms of training with us.
‘It suited us that he played at Aldershot, it helped us in terms of stature and experience – and suited him to play as well.
‘Richard has one or two leads in terms of clubs to go to, so when it has suited us he has trained with us.
‘He has not been getting in the way of anybody and we have been pleased to help him.’
Brindley featured for 76 minutes against Aldershot, before substituted with the scoreline goalless.
It meant he wasn’t on the pitch when the National League side netted four times in the final 10 minutes.