The Foxes are keen to strengthen their goalkeeping department, having failed to adequately replace Kasper Schmeichel, who joined Nice last summer.
Danny Ward has stepped up to the number-one position this term, but there are doubts over whether he can be a viable first-choice goalkeeper in the long term.
As a result, Leicester are set to delve into the transfer market this summer, and they have identified Patterson as a potential target.
The 22-year-old has featured in all 38 of Sunderland's Championship games this season, keeping 11 clean sheets.
According to The Sun, they are willing to include Iversen in a potential swap deal for Patterson if the two clubs can agree on valuations.
Iversen has spent time on loan with Oldham Athletic, OH Leuven, Rotherham United and Preston North End, but he has struggled to establish himself in the Leicester side.
All six appearances for the club have arrived this season, and he had to be patient for his first Premier League game, which arrived before the international break.
After playing 69 league matches for Preston in the second tier, Iversen would offer Sunderland plenty of Championship experience.
Earlier this month, Sunderland boss Tony Mowbray expressed his desire to keep Sunderland's best young prospects, including Patterson, but he recognised that as head coach rather than manager, some transfer decisions will be out of his hands.
As quoted by the Northern Echo, Mowbray said: "I can give an opinion on anything, but ultimately, I'm pretty sure I won't be making the decisions."
"Ultimately, for me, the proof of the pudding will be in the eating. You wouldn't want to lose Dan Neil or Anthony Patterson at 21-years-old or so. You would hope that you could find a way, contractually, to secure them.
"But if the model becomes that we're going to sell players along the way, then you hope that you get the top prices for them so that there is money to reinvest back into the team."
Patterson is under contract until June 2026, and he recently received his first call-up to the England Under-21 side.