According to James Ducker in the Telegraph, City are insistent they have made no formal bid for the French teenager having first enquired about him in April, when they "balked" at Monaco's £130 million asking price.
Indeed, while City are clearly interested in Mbappe, 18, they will not pursue him unless the fee comes down, per BBC Sport's David Ornstein:
David Ornstein @bbcsport_david
Man City categorically deny making a bid for #ASMonaco forward Kylian Mbappe. #MCFC have no intention of pursuing him at prices being quoted2017-7-26 06:36:41
Real, though, have reportedly agreed a deal in principle to sign the forward €180 million (£161 million), per Marca (via Ducker).
That has been denied, however, by Monaco, per Sky Sports News:
Sky Sports News @SkySportsNews
BREAKING::Monaco tell Sky Sports News they have not agreed a deal to sell Kylian Mbappe to Real Madrid for £160m https://t.co/DMTgUQeuEL2017-7-25 11:20:43
Mbappe's future is still hugely uncertain, but it does seem very unlikely he will be anywhere other than Monaco or Real next season.
He has become one of the most sought-after players in Europe following a breakout season in 2016-17 when he helped Monaco win Ligue 1 and reach the semi-finals of the UEFA Champions League, per football analyst David O'Brien:
Statman Dave @StatmanDave
Kylain Mbappe’s season by numbers: 2633 mins played 73 mins per goal / assist 26 goals 10 assists Wonderkid. 🌟🌟🌟 https://t.co/frmMkkHrfv2017-7-25 11:13:18
Should France international Mbappe make good on the enormous potential he showed last season and become one of the best players in the world he would arguably be worth a £160 million outlay.
But given he has only proven himself over one season in the French top flight, it would be a big risk, and it is no surprise City are not prepared to pay over the odds for him.