The goalkeeper proved a big hit with the Lilywhites in his half-season loan from Leicester City.
Iversen played 23 games after arriving in January and kept nine clean sheets.
In recent weeks the Dane boasted four clean sheets in a row, a sequence ended in the final game of the season when Nottingham Forest’s James Garner beat him from 30 yards.
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PNE are understood to have asked Leicester about taking Iversen on loan for the whole of next season.
The Foxes are open to the idea but North End might have to show some patience due to the summer plans of the two Leicester keepers ahead of Iversen in the pecking order at the King Power Stadium.
Kasper Schmeichel and Danny Ward will play at the Euros with Denmark and Wales respectively.
So it could well be that Leicester wait to see that the pair come through unscathed before letting Iversen go.
The keeper was recruited by PNE in early January when Declan Rudd suffered a knee injury. Rudd had played 22 league games on the spin before his injury, Connor Ripley having started the season’s opener as Rudd recovered from a muscular injury suffered in pre-season training.
Of the other players PNE had on loan this season, Sepp van den Berg will also be a summer target for a return to Deepdale.
The Liverpool teenager made 15 successive starts in Preston’s defence after forcing his way into the side in late February.
He came as part of the deal which took Ben Davies to Anfield.
North End have the first option on a loan for Van den Berg should the Reds want to let him out in 2021/22.
North End will consider a summer move for Liam Lindsay who ended his loan spell from Stoke by scoring the winner at Forest.
The centre-half has two years left on his contract with the Potters but had not played in their first team for 12 months before joining PNE on February 1.
Anthony Gordon and Jayson Molumby, who did not get much of a look in during their loans spells, will not be returning.
Everton winger Gordon made five starts and six sub appearances, with Molumby getting seven starts and eight games as a sub after being borrowed from Brighton.
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