The in-demand striker is now officially a free agent after running down his AFC Wimbledon contract.
Pigott scored 22 goals in 52 appearances to ensure the Wombles retained their League One status in the 2020-21 season.
The former Charlton man has unsurprisingly attracted plenty of interest as an unattached player this summer.
Pigott is on Danny Cowley's summer shopping list but, as previously reported, Pompey are braced to miss out on the marksman.
Championship duo Bristol City and Luton are keen, while third-tier rivals Sunderland are prepared to pay big wages to secure his signature.
The Black Cats are looking for a replacement for Charlie Wyke, who's been heavily linked with a move to Celtic after netting 30 times last term.
And should Pigott choose the Stadium of Light rather than Fratton Park, Kevin Phillips believes he can swiftly become a hit.
Phillips told Football Insider: ‘His stats speak for themselves. A record of nearly one in two is a great strike rate.
‘My only worry is, coming from Wimbledon, can he handle the pressure of playing for Sunderland. No disrespect to Wimbledon but Sunderland is a big step up.
‘I haven’t seen too much of Pigott but the lad can score goals, that is clear.
‘He could be the man to replace Charlie Wyke next season because it looks as if he will be moving on.
‘If Lee (Johnson) can get Pigott in and he scores a lot of goals for Sunderland he will be a hero straight away.’
Pompey are keen to bring in an additional striker this summer, with John Marquis and Ellis Harrison the only senior options.
The Blues hoped to prise Jayden Stockley back to the south coast. However, he instead decided to return to Charlton Athletic after spending the second half of last season on loan at The Valley.