City have already signed Bernardo Silva for £43 million and are on the verge of wrapping up their second signing of the summer with the £35m acquisition of Benfica goalkeeper Ederson.
Yet they are keen to press ahead with the rest of their business and director of football Txiki Begiristain travelled to London shortly after the deal for Silva was closed at the end of last week.
It has been made clear to City that Arsenal would prefer to sell Alexis to Bayern, having resigned themselves to the fact the Chilean will leave this summer.
Relations between Alexis and his Gunners team-mates have grown increasingly strained in recent weeks, and Bayern are hoping to take advantage of the situation by matching Arsenal's asking price and offering him a lucrative wage packet.
But City have long known the former Barcelona forward wants to link up with Pep Guardiola and are determined to make it happen by at least matching Bayern's transfer offer and trumping any wage packet that the forward could receive elsewhere.
Although City moved quickly to sign Silva and Ederson and are hopeful of finalising other deals in the coming weeks, they are prepared to wait for Alexis given Arsenal will try everything to move him abroad.
As well as holding talks over a move for Alexis, Begiristain also attempted to broker a deal for Tottenham right-back Kyle Walker at the weekend.
Despite failing to make any significant headway for the England international — with Spurs playing hardball over the fee — a source close to the negotiations informed Goal on Monday that the Blues are confident of landing Walker and the Arsenal forward, as well as Monaco left-back Benjamin Mendy.
Goal sources in France say Mendy, 22, has agreed to join City, though it is understood the finer points of a deal are still being worked out with Monaco as other top European clubs, including Juventus, also vie for his signature.
Mendy sparked a brief flurry of excitement on Sunday evening when he posted a picture from a private plane, only to cool speculation when he landed in Paris shortly afterwards.
Guardiola has identified Mendy, Walker and Alexis as vital to his plans for next season and those familiar with negotiations believe City will sign all three, even if Alexis arrives later this summer.
That would not conclude City's business, however, as sources close to the City hierarchy have indicated Guardiola is keen to add between six and eight new faces in total.
The Blues could yet bring in another forward even if they sign Alexis, with Guardiola regarding new arrival Silva as predominantly a central playmaker rather than a wide option.
Though Silva played much of last season on the right wing for Monaco, Guardiola intends to rotate the Portugal international with Kevin De Bruyne and David Silva. In fact, De Bruyne is currently regarded as a more viable option for the right-hand side than Bernardo Silva, given the Belgian's direct style and crossing ability.
Guardiola wants that kind of flexibility in his team and plans to have three players — the two Silvas and De Bruyne — to cover two central attacking midfield roles. Ilkay Gundogan will provide an additional option for the advanced and deep midfield positions when he fully recovers from a serious knee injury.
Bernardo Silva is likely to play some games out wide as Guardiola rotates his team next season, but the Catalan does not see him as a similar player to Leroy Sane, Raheem Sterling or Gabriel Jesus.
Guardiola is planning to use three speedy forwards next season — either two wingers and a No.9, or two No.9s and a winger — and wants five or six different players to cover the three positions.
The City boss wants enough options to keep all of his players fresh for an assault on all four competitions, having finished last season trophyless for the first time in his managerial career.
Guardiola hopes to have nine top-class players — the two Silvas, De Bruyne, Sane, Sterling, Jesus, Alexis, Sergio Aguero and potentially another pacy winger capable of scoring goals — for five positions.
Guardiola also sees Yaya Toure as a valuable option at the back of midfield, and Goal understands the Ivorian has agreed to stay at the Etihad Stadium for one more season.
City are also keen on Leonardo Bonucci, though the centre-back's future will not become clear until after next Saturday's Champions League final. The Italian was extremely close to signing for the Blues last summer but elected to stay at Juventus as a result of his son falling ill.
Guardiola believes Bonucci and Vincent Kompany would bring the best out of John Stones, who could take over Pablo Zabaleta's No.5 shirt next season.