The 37-year-old is currently a free agent having ended his five-year stint with the Los Angeles Galaxy towards the end of last year, and has since been linked with a move to a whole host of clubs.
As in the past, Beckham will train with North London outfit this week in order to maintain his fitness as he mulls over a decision on his future.
However, Beckham, who has also been the subject of interest from West Ham United and Queens Park Rangers, could be about to join the Gunners in a sensational move at the end of the January transfer window.
A source close to the newspaper said: "Beckham wanted this all kept hush-hush. You have to wonder why?"
The ex-Manchester United winger has previously enjoyed spells at Real Madrid, AC Milan and Preston North End.