The manager had previously mentioned that the 19-year-old Brazilian has been out of action since joining the Eagles with a ‘stress’ injury, and has been awaiting a second scan.
Speaking at his pre-Brentford press conference, Hodgson expanded on this, explaining: "There isn’t really a timeframe [for França] because he will need another scan to start with, and that will then tell us if the injury that he had when he joined us is progressing and clearing up.
“Then, after that scan – which will hopefully take place before the end of the month – then we’ll draw up a new plan.
“In the very best of scenarios it will be that he’s okay, and we can really accelerate his training and get him back onto the field of play with us. In a worst case scenario, it might mean he needs more time to recover from the injury.”
Hodgson also provided an update on the fitness status of Will Hughes, who was injured against Millonarios in Chicago in pre-season.
“He’s getting better,” the manager said of the midfielder. “He’s making very good progress. It was a nasty injury.
“We’re hoping because he’s progressed so well, we might see him back with us in the middle of next week. He has to first of all pass through certain procedures so that the medical and sports science people can clear him to join us.
"The latest I heard is that they are expecting him to be available to train with us on Thursday of next week."
With França, Hughes and Michael Olise – who last week put pen-to-paper on a new four-year deal with the club – all currently recovering, Hodgson reaffirmed he has been happy with Palace’s performances despite their absence.
The manager noted: “It’s always unpleasant, I suppose, to have players in your squad you can’t use, but in terms of the two matches we’ve played, I think we’ve dealt with their absence very, very well.
“I’ve been more than satisfied with both of the teams’ performances [against Sheffield United and Arsenal], but I shall also be very pleased when those three make themselves available again because that will strengthen our attacking positions, which at the moment we are somewhat limited in.”
Hodgson also confirmed he expects Jean-Philippe Mateta to be available for selection against Brentford, having missed Monday's match with a knee problem.
"He's on the bench tomorrow," the manager observed. "There's a good chance we will see him during the course of the game."