Having had a £50m bid rejected on Tuesday, Goal understands City submitted another offer worth around £60m on Wednesday evening, a figure the Gunners approved on Thursday morning.
But the London club will only sign-off on the deal if they can recruit another forward in the remaining hours of this summer's transfer window.
That is now the main obstacle that stands in the way of City finally securing Sanchez's services, having coveted the Chile international for several months.
Arsenal had been keen to include Raheem Sterling in the deal but City made it clear from the outset that would not be an option.
Goal understands Arsenal have reignited their interest in Monaco winger Thomas Lemar, as they believe the money they would receive from the Sanchez deal would allow them to tempt Monaco into selling.
Julian Draxler has also been mooted as a potential replacement for Alexis but Goal understands that move will not happen and that Lemar is the club's first-choice target at this late stage of the window.
Arsenal’s struggle to add to the signings of Alexandre Lacazette and Sead Kolasinac this summer is partly because of their failure to offload fringe players on high salaries.
The Gunners can only increase the wage bill by £7m above what it was last season after the Premier League Short Term Cost Control (STCC) rules came into play from the new campaign.
They are looking to send the likes of Lucas Perez and Joel Campbell on loan in a bid to meet the new rules, which only allow for overspend if a club can show that it comes as a result of increased commercial or matchday revenue, or player sales.
Arsene Wenger’s side had a wage bill of £191m two seasons ago, according to the most recent club accounts available via Companies House, and therefore would need to sell Alexis before bringing in a high-profile player.
City are desperate to sign Alexis and it is not yet known what course of action the Blues would take should Arsenal fail to sign a potential replacement and therefore refuse to sell.