The 23-year-old France left-back, who joined Monaco from Marseille last summer, played 34 times for the club last season and helped them to their first Ligue 1 title in 17 years.
His fee would eclipse the initial £45m paid by City for right-back Kyle Walker earlier this month and the £47.5m deal for centre-back John Stones last summer.
City have not yet confirmed the deal.
Mendy would become the Premier League club’s fourth major signing of the summer and take their spending to more than £175m, following the recruitment of England international Walker, Portugal midfielder Bernardo Silva (£43m) and Brazilian keeper Ederson Moraes (£35m).
City are also close to signing Real Madrid’s 26-year-old Brazil right-back Danilo. They have sold left-back Aleksandar Kolarov, 31, to Roma for £4.5m.
Before Thursday’s pre-season game against Manchester United in Houston, manager Pep Guardiola said the club were talking to “three or four” more players.
“They are young players, with the next three, four, five years in mind,” the Spaniard added.
“We have options for players coming but until the deal is done, out of respect for the other clubs, I cannot say anything.”
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