According to the Mirror's John Cross, the Gunners are in ongoing negotiations with the Chile international, whose camp have also held talks with Bayern, while City are said to be "confident" of snapping him up.
The 28-year-old has little more than a year remaining on his contract at the Emirates Stadium, and Arsenal are adamant he will not be leaving this summer.
It's imperative the Gunners keep Sanchez if they are to challenge for the Premier League title. The forward has plundered an incredible 29 goals in all competitions this season, as well as producing 19 assists in all competitions.
ESPN FC's Mattias Karen noted his linking play with Mesut Ozil in the former match, while Bleacher Report's James McNicholas hailed him after scoring a brace in the latter:
His efforts have put him among the top performing strikers in European football this year, according to the International Centre for Sports Studies' Football Observatory:
Arsenal's task of keeping him has been made even harder by missing out on the UEFA Champions League, but the reported deal would put him among the Premier League's top earners. The Gunners will have to hope it's enough to persuade him to stay.
Meanwhile, according to Bild (h/t the Mirror, via Birmingham Mail's Steve Wollaston), Arsenal are reportedly interested in Thorgan Hazard, whom Borussia Monchengladbach have put a £10 million price tag on.
The 24-year-old, younger brother of Chelsea star Eden, had a strong campaign and racked up 11 goals and seven assists in all competitions.
Hazard is a versatile operator and has spent time on both flanks, at No. 10 and up front this season.
Here's a look at some of his highlights:
If he's available at the reported asking price, he could make for a useful addition to Arsenal's squad.
However, he's not a player who significantly upgrades their starting lineup, if at all—there are more pressing priorities for the club to address in relation to their playing personnel.