The Chilean did not travel for the game at the Vitality Stadium, with the suggestion being that his head has been turned by interest from Manchester United.
Arsenal boss Arsene Wenger admitted for the first time on Friday that it was looking likely that his star man would leave the club when his contract expires at the end of the season.
It had previously looked like the 29-year-old was destined for a reunion with former Barcelona manager Pep Guardiola at Manchester City, who saw a deadline day bid for the forward fall through in the summer after Arsenal failed to bring in a replacement.
However, after City refused to pay Arsenal's £35m asking price, United are thought to have stolen a march on their rivals by offering a deal involving playmaker Henrikh Mkhitaryan in part-exchange.
Arsenal are thought to have been keen on the Armenian when he left previous club Borussia Dortmund in 2016, only for him to opt for a move to Old Trafford.
Speaking ahead of the game against Bournemouth, Wenger confirmed that Sanchez's future was the reason for his lack of involvement and said that his future was set to be decided.
"He’s been vague at the moment, his situation is not decided one way or the other so I left him at home," said the Arsenal boss.
"Yes, but don’t read too much into it because even I don’t know what way it will go," he explained after being asked if a resolution was imminent.
English publication 'The Daily Mail' reports that Sanchez's agent has recently flown into England, suggesting that his future may be set to be resolved in the coming days.
With United prepared to blow City out of the water when it comes to personal terms, could Jose Mourinho's side pull off the shock of the transfer window?