Qualifiers away to Malta and at home against Slovakia could see England all but secure their place at Russia 2018 if they secure two victories.
This weekend's second round of Premier League fixtures will be the last opportunity to impress Southgate as the squad will then be named before next weekend's games.
Having made an early start to their competitive campaigns, the Everton trio of Jordan Pickford, Michael Keane and Wayne Rooney have all impressed and were watched at Goodison Park by the England manager on Thursday evening as the Toffees defeated Hadjuk Split in the Europa League.
Across Merseyside, Adam Lallana will miss out on the games after picking up an injury in pre-season that has ruled him out of action for the opening two months.