"I can tell you that he is focused, and my decision is clear: he will stay and he will respect that".
The Chilean was in fantastic form for the Gunners last season, scoring 30 goals and assisting a further 19 strikes in all competitions.
But Spanish newspaper, Mundo Deportivo, reports that Arsene Wenger has entered the race to sign the France forward and an offer has been rejected by the Bundesliga club.
Sanchez has previously admitted he wants to play in the Champions League, something Arsenal can not offer this season, but if the report is true it seems he may have accepted he will have to see out his contract. If the 28-year-old hasn't committed his future to the Gunners by January, he'll be free to sign a pre-contract and join his new club for the 2017-18 campaign. The transparency of society is very pushy but we have to resist a little bit, as much as we can.
Wenger said there was "nothing to announce" regarding the futures of Mesut Ozil, Alex Oxlade-Chamberlain, Jack Wilshere and Kieran Gibbs, who are also out of contract next year.
Giroud is said to be Everton's priority target, but should the Frenchman decide to stay at Arsenal the Toffees will then turn their attention to Welbeck.
Petr Cech hailed Arsenal's squad strength after their FA Community Shield shootout victory over Chelsea at Wembley.
"We have a congestion of strikers", Wenger added.
He added: "The players we have brought in are quality and in a very difficult global market we've worked hard all summer and I must say the competition is so difficult in market it's hard to find players who can strengthen and have the mental attitude". "We want to find a good balance between the confidence we can get out of this game and the urgency that every Premier League game demands", he said.
Sanchez was running out of options for a summer move, with Arsenal adamant that they wouldn't sell to a Premier League rival.