Sterling rejoined the England World Cup squad at their Al Wakrah base on Thursday, and the Chelsea forward was expected to be involved in training on Friday afternoon.
Sterling flew back to the UK last Sunday following a break-in at his family home and missed the 3-0 last-16 win over Senegal on the same day.
Southgate had never previously started a knockout game without Sterling, who turned 28 yesterday, but Phil Foden and Bukayo Saka impressed in his absence and are set to keep their places in the quarter-final against the world champions.
Sterling scored in the 6-2 win over Iran in England’s opening group game and kept his place for the subsequent draw with the USA, before being rested against Wales.
Asked if Sterling could play a part against France, Southgate told ITV: “We will have to assess that. He has missed a lot of training and had two long flights. That is not good preparation for a game of this standard, but let’s see how he is.”
If the 81-cap forward is not deemed fit enough to be involved, Southgate will still have no shortage of attacking options, with Marcus Rashford and Jack Grealish likely to be in reserve and James Maddison pushing for his first minutes of the tournament.
Sterling was left shaken and concerned for the wellbeing of his fiancee and young children after intruders entered his Surrey home.
Police have said they are investigating a report of a burglary, after the occupants of the house returned to find jewellery and watches had been taken, but that no threats of violence were involved.
Southgate said: “[Raheem] has felt the need to support his family and to be there for them — and that is a simple decision for me: I have to support the player. I remember Fabian Delph going home, his wife was expecting, in Russia [during the last World Cup]. We did the same thing. There are moments for every individual where family becomes most important. I’m always supportive of that.
“Of course I want the best players available and want to win matches, but as a manager you have to recognise there are certain moments in people’s lives that are more important at that particular time.”
Saka has scored three goals in as many starts so far, with a brace against Iran in the group stage before sealing England’s 3-0 victory over Senegal in the last-16.
England training - Behind the scenes
The Arsenal winger is dreaming of going one better than at Euro 2020, and his team-mate Aaron Ramsdale believes the 21-year-old is showing he belongs on the biggest stage.
“I don’t think Bukayo needed to prove it too much anyway,” said the goalkeeper. “We all know what he can do and how he goes about things — it was just actually doing it in a World Cup game. He has done that now. He has scored three goals and is playing really well.
“We have just got to carry on as a team helping him and helping each other to get us to where we want to be.”
Declan Rice is fit to play, after the West Ham midfielder returned to training on Thursday following illness. Callum Wilson is also expected to be fit, despite struggling this week with what was described as a minor strain.