The Premier League outfit are keen on the Pompey loanee, who could potentially succeed England’s No1 Jordan Pickford at Goodison Park.
Frank Lampard is said to be ready to listen to offers for Pickford this summer, with the Toffees hopeful of securing their Premier League place after an upturn in form.
Whatever happens there, Everton are said to now be among the swathe of top flight and Championship clubs eyeing a move for the 20-year-old - according to the Irish Independent.
Southampton, Liverpool, Bournemouth, Sheffield United and Preston North End are others said to be keenly trailing Bazunu.
Everton have been watching the keeper’s excellent progress at Fratton Park this term, in a season which has seen him grow and establish himself as Republic of Ireland’s first-choice keeper.
Manchester City are said to be deciding whether to let the former Shamrock Rovers man leave on loan again or depart permanently, with two years remaining on his existing agreement at the Etihad Stadium.
Any permanent City exit for Bazunu, whether at home or abroad, will likely include a buy-back clause in favour of the Premier League champions.
And the prospect of regular first-team will be a motivating factor in his decision.
Bazunu made that clear last month when he spoke of his appreciation for Pompey, and the springboard they’ve offered him to develop his career.
The impact he’s made was underlined when he received the appreciation of his peers, after being named Pompey players’ player of the season at the club’s end-of-season awards at the start of the month.
Bazunu has made 46 appearances this season for Pompey, with a further six outings arriving for his country.
In that time it’s not just been his ability to keep the ball out of the net which has come to the fore, but also his talent at promoting attacking play with his ball-playing skills.
When speaking about his future this month, Bazunu told Pompey’s YouTube channel: ‘I have absolutely no idea (what will happen) but I'm open to everything.
‘I've got to go and meet some people at Manchester City and see what they have to say, but my aim is just to go and play again. I've got to continue to get games under my belt and keep these experiences coming.’