It is likely that the Premier League champions will bring in just two new players ahead of next season, but they have long been considering a big-name, big-money signing for the forward line and it is believed that they would still be open to such a deal even if Mahrez, or Leon Bailey, is signed before the start of pre-season in July.
But Mahrez, who attempted to force a move to City in January, is currently the most likely option. It is understood that talks are ongoing with Leicester, although the two clubs are some way apart in their respective valuations of the Algeria international.
It is a similar situation regarding Italy midfielder Jorginho. Sources have told Goal that the Blues have already seen €45 million bid rejected for the 26-year-old, although that is not regarded by club insiders as any kind of set-back and the two parties are continuing to work on a deal.
Jorginho's agent is set to meet Napoli officials next week to help reach a conclusion. City remain confident they will bring him to the club, having agreed personal terms some time ago, but there is a degree of concern that a rival bid from Chelsea could turn the midfielder's head if Maurizio Sarri, who has just left Napoli, takes over from Antonio Conte.
City are keen to wrap up a deal as quickly as possible, having prioritised Jorginho over Shakhtar Donetsk midfielder Fred, who they pursued throughout January.
Goal revealed earlier in May that the Premier League champions had established two separate short-lists of candidates to strengthen their attack.
There are two names on each shortlst; Mahrez and Bailey are the comparatively cheaper, more attainable options, while Paris Saint-Germain's Kylian Mbappe and Chelsea's Eden Hazard are regarded as more ambitious, marquee targets.
While City have pressed ahead with moves for Mahrez, with Bailey in reserve, it is understood that has not necessarily brought an end to their pursuit of a more expensive signing.
City insiders still believe there is some chance, however remote, that either Mbappe or Hazard could become available later this summer. Both players will be involved in the World Cup and both would prove extremely difficult to prise from their clubs, so the Blues are ready to move ahead with other targets and assess the lay of the land closer to the start of next season.
There would be many obstacles to overcome if either player were to be signed. Neymar's situation at PSG and the French club's finances are among the factors that will affect Mbappe's future, while Chelsea's next manager and Roman Abramovich's continued investment in the club would have a bearing on Hazard's.
It also remains to be seen whether City would have to sell a first-team player to bring in a player worth upwards of £100m.
Pep Guardiola insists he is looking to bring in one or two new faces and that the club are not in a position to invest a record transfer spend this summer, but sources say Mbappe or Hazard would be signed if everything comes together.
“Yaya [Toure] is leaving so we have to replace him, and maybe another one,” Guardiola said on Sky Sports’ Monday Night Football earlier this month. “One or two more, no more. Maybe people don’t believe me, but we don’t have the money to invest £300m every season."
Any big-money signing would only be considered a bonus by Guardiola, who considers Jorginho and Mahrez or Bailey as priorities.
Guardiola wants those two transfers to be completed quickly so they can join the club's pre-season tour of the United States and be settled into the squad for the start of next season.
Mahrez, Bailey and Jorginho are not going to the World Cup. That is no coincidence; Guardiola hopes his new signings can bring extra stability to the squad heading into next season, given the Blues are providing 17 players for the tournament in Russia, more than any other club.
It is believed that another of the main considerations relating to City's priority targets is their willingness to fight for a place in the first team, and accept time on the bench. Having spoken to Mahrez at several junctures since January, and Jorginho in recent months, City are convinced that their targets have the right mentality to fit into a happy dressing room environment.
Senior City sources also stress that a centre-back is not on the agenda this summer. It is believed that the club are keeping tabs on Ajax's Matthijs de Ligt,18, but it is currently regarded as unlikely that he would be signed ahead of next season.
Only serious rival interest in the youngster, or an injury to a current City centre-back, could force the club to step up their interest.