Pep Guardiola has been forced to field Fernandinho in the role since losing Aymeric Laporte to serious injury early in the season.
The form of the Foxes duo has caught the attention of the Etihad club, according to Sky Sports News.
City are considering several centre-back options after failing to replace the outgoing Vincent Kompany in the summer before struggling with injuries.
They have fallen nine points behind Liverpool in the title race, but Guardiola has previously insisted that the club are not looking at bringing in reinforcements during the January transfer window.
Leicester have the best defensive record in the Premier League this season, conceding just eight goals in 12 matches.
That form has helped them storm to second place in the standings ahead of the international break.
Former Leicester defender Harry Maguire was a target for the Citizens over the summer before he joined Manchester United, but Brendan Rodgers’ side have coped admirably in his absence thanks to the form of Evans and Soyuncu.
Evans came close to joining City while he was at West Brom in 2017, as the Baggies rejected several bids before he eventually joined Leicester for £3.5 million.
The Northern Ireland international has played every minute of Leicester’s Premier League campaign so far alongside Soyuncu.
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