The East Anglian Daily Times claim Danny Cowley is interested in the 24-year-old, who came through the youth system at Portman Road.
The same report says that Lincoln are leading the chase for Bishop, who has been told by Paul Cook that he can leave the club.
MK Dons and newly-promoted Scottish Premiership side Hearts are also keen.
But it has been suggested that the Blues could jump to the head of the queue if a swap involving fellow attacking-midfielder Michael Jacobs could be worked out.
The News revealed last month that former Wigan ace Jacobs had been given permission to speak to Ipswich, with the 29-year-old a pivotal player under Cook during the Latics’ League One title win in 2017-18.
However, he will not be allowed to leave Fratton Park unless it is on Pompey’s terms, with Cowley a big admirer of Jacobs, regardless of his injury record during his time with the Blues.
It’s feasible that a swap involving Bishop could appeal, with the Blues keen to reinforce their attacking options.
At present, it’s the only department Cowley has been unable to strengthen this summer, following the collapse of a move for forward Jayden Stockley.
Despite Pompey’s League One rivals tiggering a clause in Bishop’s contract at the end of last season, he is currently training with the under-23s at Ipswich alongside Kayden Jackson, Myles Kenlock and Flynn Downes.
All four are expected to leave Portman Road this summer as Cook carries out a major overhaul of his playing squad – a rebuild that has already seen 20 senior players depart.
Bishop has played 132 times for the Tractor Boys since making his debut in 2014.
Last season he made 38 appearances, scoring four goals.