Turkish journalist Ekrem Konur reports the Old Trafford club have identified Soyuncu as a target.
Talks are not yet underway but the report suggests United hope to begin discussions imminently.
Konur adds Chelsea continue to monitor the situation, too.
Soyuncu, 25, joined Leicester from Freiburg for £19million in 2018.
Soyuncu enjoyed a stellar 2019-20 campaign and was named in the PFA Team of the Year.
Despite struggling with injuries and Wesley Fofana's emergence, the centre-back still racked up 32 appearances last season, including the full 90 minutes as City beat Chelsea in the FA Cup final.