Spanish newspaper El Mundo Deportivo say the deal is all but complete, with personal terms between the Chilean and Arsenal agreed. All that remains to be done is the medical, after which the deal will be announced.
The Spanish newspaper also carry a picture of Sanchez at an airport seemingly getting ready to jet into London in order to sign for Arsenal - although there's no confirmation that the image is recent.
Arsenal are believed to have paid around £30 million upfront, with the deal rising to £35 million - or €40 million - with add-ons. BBC Sport's David Ornstein also reports that the transfer is close to complete, but the Barcelona winger won't be unveiled in time for Arsenal's new kit launch next week.
Sanchez joined Barcelona in 2011 from Udinese for a fee of £23 million and made 88 league appearances for the Catalan club. He scored 19 goals last season but couldn't help Barcelona claim a single piece of silverware, and is now being moved on as part of a sizeable squad overhaul at the Camp Nou.
The deal will be Arsenal's first of the summer as they look to add to their squad in order to challenge for the Premier League title next season - and there should be plenty more to follow if reports are to be believed.
An £11 million deal for Mathieu Debuchy is said to be close to being complete, while Loic Remy could also follow the pair to north London. The Gunners finished fourth last season and ended their eight-year trophy drought by claiming the FA Cup with victory in the final against Hull - but are now looking to kick on to bigger and better things.
The deals for Sanchez, Debuchy and Remy are expected to cost around £50 million, and reports suggest more spending will follow.
Arsene Wenger is said to be keen on Bayer Leverkusen's Lars Bender, while he is also in the market for a new forward. As yet it remains unclear who the man for the job could be, but one it certainly won't is Mario Mandzukic, who has signed on the dotted line for Atletico Madrid.