The Gunners had hoped to persuade Alexis to extend his current contract, which expires next summer, or at least to sell him abroad as a last resort.
But sources close to the situation have informed Goal that the Gunners are now deeply concerned they will lose the Chile international to City following talks with his representatives.
Alexis' future had been on hold while he participated in the Confederations Cup, though Arsenal had prepared a huge pay rise in an attempt to keep him at the Emirates Stadium.
The Gunners now believe there is little chance of him even listening to their offer.
Paris Saint-Germain and Bayern Munich had explored the possibility of signing the 28-year-old in recent months but he wants to be reunited with Pep Guardiola, who coached him for one season at Barcelona. Any rumoured move by Inter, therefore, is destined for failure.
City have long been confident they will eventually get their man and it is now thought unlikely that the Gunners will hold Alexis to the final year of his contract and lose him on a free next summer.
Alexis insisted last Friday that he had made a decision on his future, while his international colleague and City goalkeeper Claudio Bravo made it clear that a transfer will be discussed.
"Of course I would love to have him as a team-mate," he said. "But it's not a matter just for the two of us. It's something the clubs have to go through."
City have not yet made a formal bid but Arsenal are now preparing themselves for an official approach in the coming weeks.
Guardiola has made Alexis his top priority this summer and the Blues, having initially been happy to bide their time, will want to secure his signature as soon as possible.
A bid of £50 million may be necessary to persuade Arsenal to sell, albeit reluctantly.
Sources close to the City hierarchy are concerned Arsene Wenger will delay the deal for as long as possible, perhaps with a view to depriving Guardiola of the Chilean's services for the opening games of the Premier League season.
Though this saga is set to run for several more weeks, City are ready to officially announce the signing of Dani Alves.
The Brazil right-back agreed a two-year deal with City over a fortnight ago but Guardiola was happy for him to enjoy his holidy before flying into Manchester to put pen to paper.
City are back in pre-season training on Monday and they expect to have Alves on board before their fly to the United States for the International Champions Cup on July 17.
The Blues are still working hard to tie up deals for full-backs Benjamin Mendy, Kyle Walker and Ryan Bertrand, as well as a centre-back.