Sanchez, who is the subject of a January move to either Manchester City or Manchester United, has six months left on his contract and Gunners manager Arsene Wenger appears ready to sell Chilean in this transfer window, rather than losing him for nothing in the summer.
"He's being vague at the moment".
But while Wenger confirmed he expected the situation to be resolved soon, he said he did not know what the final outcome would be.
Despite dictating much of the early possession Arsenal clearly missed their more illustrious attacking talents with Alexandre Lacazette cutting a lonely figure without the cutting edge of Sanchez and the injured Mesut Ozil.
Arsene Wenger: 'Alexis Sanchez decision imminent'
Asked if Sanchez's exit is imminent, Wenger said: 'Yes, don't read too much in it because even I don't know which way it will go. United have a lot of creative players in the final third, but none is as influential as Sanchez.
But after Arsenal's opener Bournemouth mounted pressure and nearly restored parity through Ibe who fired a tame shot which was saved by Petr Cech.
In a goalless first period, Ainsley Maitland-Niles came closest for Arsenal, as he rattled the bar. In the 35th minute, Danny Welbeck's shot was saved by keeper Asmir Begovic but if the Arsenal striker had gone to ground under the challenge of Simon Francis, he may well have won a penalty.
On the pitch, however, the Gunners could not lift the gloom that has surrounded the club in recent weeks, despite Hector Bellerin's cross-shot somewhat fortuitously giving the visitors a lead in the 52nd minute. Bellerin ran onto an excellently weighted pass from Alex Iwobi to steer the ball into the Bournemouth net.
But the Cherries' response was fantastic, and two goals in four minutes from Callum Wilson and Jordon Ibe put them 2-1 ahead and on course for a famous win.