City made an initial bid of £50m for the Algerian on Tuesday, but Leicester were quick to reject that offer out of hand.
Pep Guardiola is believed to see Mahrez as the perfect man to offer cover for the injured Leroy Sane out wide, but in order to bring the 26-year-old in, it looks like City will have to break their transfer record for the second time in 24 hours.
The Premier League leaders returned to Leicester with a fresh £55m bid according to 'The Daily Mail', only for that to once again rejected, with Mahrez's current employers valuing him well in excess of £60m.
Any deal approaching that figure would exceed the £57m City paid Athletic Bilbao for defender Aymeric Laporte on Monday, though at this moment in time Leicester are thought to have no intention of selling.
That stance should change though if City significantly up their offer and the player makes his feelings known by refusing to play against Everton this evening.
Mahrez is known to be keen on the move, having slapped in a transfer request after hearing of City's interest.