The attacking midfielder enjoyed a successful loan spell at Doncaster Rovers last season.
The Champions League holders have again earmarked the 20-year-old for loan experience during the current campaign.
It is understood that scenario has brought Kane to the attention of the Blues, with the League One and League Two transfer window open until September 2.
The former England under-17 international has been tipped for a bright future at Anfield, signing a new deal in December 2018.
Last term’s initial six-month stay at Doncaster developed into a season-long presence, making 49 appearances and scoring seven times.
Kenny Jackett has this summer recruited eight players to bolster Pompey’s promotion aspirations.
And, following yesterday’s closure of the Premier League and Championship transfer window, Kane is available – and in the Blues’ sights.