According to website Football Insider, the Blues are lining up a move for the ex-Tottenham defender.
Ogilvie completed a permanent switch to Priestfield last summer, after spending the 2018-19 season on loan from Spurs.
He impressed for the Gills last term, scoring four goals in 41 appearances before the campaign was curtailed amid the Covid-19 crisis.
That led to the 24-year-old being awarded Steve Evans' side's Player of the Season prize.
Ogilvie's one of the Gillingham's prized assets and is also capable of playing centre-back.
He has a year left on his current deal, meaning Pompey would have to pay a fee for the former England under-17 international if they are interested.
Certainly, Ogilvie's versatility is something Kenny Jackett is hunting in the summer transfer window.
The Blues are keen to bring in a left-back and a left-footed centre-half – meaning Ogilvie could fill both roles.
Lee Brown is currently the only senior option at the club, after Brandon Haunstrup left for Kilmarnock and Steve Seddon returned to Birmingham following his loan spell.
During the lockdown period, The News pinpointed Ogilvie as someone Pompey could be tempted to lodge a bid for this summer.
As reported earlier today, Jackett is also targeting Ogilvie's Gillingham team-mate Brandon Hanlan as a potential forward addition.