But the Blues know they will likely have to hold their nerve to bring in the man who finished as their top scorer this season.
Ipswich are the latest name to be linked with a bid to bring in the 23-year-old, after a powerful second half to his loan from Leicester City.
And there is Championship interest in Hirst, with his form since hitting his first Pompey goal in November catching the eyes of suitors from a higher level.
Pompey would likely have to make an increased contribution to Hirst’s wages from what they paid this season.
That commitment would be one of the biggest of any of the players in Cowley’s squad next term.
But that is something Pompey are prepared to do to get the man their boss has identified as crucial to his squad plans moving forward.
The likelihood, however, is Cowley is going to have to wait until later in the summer window to get a deal over the line.
Leicester boss Brendan Rodgers has indicated he wants to run the rule over Hirst, when his players return to pre-season training.
That is standard practice for most managers, with those at a higher level tending to hold on to their younger players until their own squad planning crystalises.
That will still create a wait which will probably income increasingly nervous for Pompey fans.
With the Blues likely to fund only a loan deal, there is the possibility of rivals trying to engineer a permanent move for Hirst.
Cowley will hope the lure of regular first-team football at a club where Hirst became a popular figure will work in his favour, however.