The 28-year-old did not feature in Sunday's last-16 win against Senegal due to concerns over his family's wellbeing following an incident at his home in Surrey.
Sterling flew back to the UK to deal with the matter but gave the green light to the Football Association to look at the logistics of bringing him back before Saturday's crunch clash.
He will train with his team-mates at their Al Wakrah base on the eve of the game but Southgate believes the week could take its toll on Sterling.
"We will have to assess that," the England boss told ITV Sport when asked if Sterling could feature against France.
"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."
Sources close to the player initially indicated armed intruders had broken into his home while his family were inside, but subsequent police statements said the occupants of the house had reported items missing after returning to the family home from an international trip.
Police said on Monday they were working to establish precisely when the break-in took place, and also offered assurances to the local public that no witness statements had been made regarding the involvement of armed suspects or threats of violence.
Surrey Police issued a statement on Wednesday morning saying they were working to establish whether the arrest of two men on Tuesday night in an area close to where Sterling lives had any connection to the burglary at his home, which was reported on Saturday.
That is understood to have brought initial peace of mind to Sterling, although an update on Wednesday afternoon said "at this time" the arrests were not being linked to the break-in at his house.
"He has felt the need to support his family and to be there for his family and that is a simple decision for me - I have to support the player," Southgate said.
"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 football 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."