The 20-year-old spent the first half of last season on loan with Northampton Town, making his professional debut with the Cobblers in League One.
Thompson played the full 90 minutes in the Under-21s' 1-1 draw against Sunderland last week and now makes the move to newly-promoted Chesterfield following their return to the EFL.
The Spireites currently sit ninth in League Two after a win and two draws in the early part of the new campaign.
Everyone at Newcastle United would like to wish Max well for his loan spell.