The 21-year-old has gone away with the Blues ahead of a potential summer switch, having yet to pen a new deal with Swindon.
Odimayo has been linked with Championship trio Stoke Derby and Preston – along with League One rivals Sheffield Wednesday and Doncaster – in recent months.
However, he bids to win a Fratton Park deal, with the centre-back opting not to pen a new deal at the County Ground despite being offered fresh terms.
Joining Swindon last summer following his release from Reading, Odimayo enjoyed a breakthrough season in the 2020-21 campaign.
Although he couldn’t prevent the Robins from suffering League One relegation, he was one of their stand-out performers and made 33 appearances in all competitions.
During his time at Reading, Odimayo made one outing in the Carabao Cup and had loan spells with non-league Hungerford Town and League of Ireland outfit Waterford.
The head coach has signed Clark Robertson on a free transfer from Rotherham, while Paul Downing and Sean Raggett both have a year remaining on their respective deals.
However, Cowley covets an extra body as he plots Pompey’s latest promotion push.
Norwich striker Gassan Ahadme has travelled after his hat-trick in the 5-2 win at the Hawks, along with ex-Southampton midfielder Jake Hesketh.