Portugal was one of the favourites to win the competition in Qatar but was knocked out by Morocco in the quarter-finals.
Santos has led the team successfully to the top of their World Cup group before hammering Switzerland 6-1 in the round of 16.
Portugal won two of their group games as they beat Ghana and Uruguay, although they did lose to South Korea.
The manager had enjoyed success with the national side as he helped them to the trophy at Euro 2016, just two years after he was appointed. Portugal also won the 2019 Nations League finals under Santos.
The qualifying campaign for Euro 2024 starts on March 23 next year, as Portugal takes on Liechtenstein.
It has been reported that former Chelsea and Manchester United manager Jose Mourinho is the Portugal chiefs’ favourite to take the job.
The Portuguese FA would like to have the current Roma boss in position well before the next campaign.
This allows him time to identify Portugal’s top talents for a new tactical system.