The midfielder has been at London Colney this week with Under-23 boss Steve Bould who will continue to assess his potential.
The 22-year-old was released from Almere City over the summer, a team which operates in the second tier of Dutch football.
Bergkamp came through the youth system at Almere City but never made a first team appearance for the side.
Mitchel grew up in London, born in 1998, with his father playing for the Gunners from 1995-2006, scoring 120 goals in 423 appearances.
The youngster left London when he was 14, but maintains a strong connection with the city and Arsenal.
‘My father also played for Internazionale and Ajax, two wonderful clubs, but Arsenal feels like home,’ he told ELF Voetbal last year.
‘Whether I choose the Emirates or Highbury? I would have preferred to play at Highbury. There are the memories of everything my father has done for the club.’
‘When the time is right, I’d love to go back and get a role somewhere,’ he told FourFourTwo in August.
‘At the moment we’re great in Holland. I’ve got a young family and have been out of coaching for a little while now.
‘But I must admit, I have the urge to go on the pitch again and help, whether it’s with the technical or coaching staff.
‘It might be a bit too early, but you never know. We’ll see what comes my way.
‘At the moment there hasn’t been any contact about that, but as soon as I make my decision, let’s see what happens.
‘Of course it would be ideal at Arsenal – I spent 11 years there and have a good feeling with the club.’