The world number one withdrew from the Miami Open after suffering the injury in training, with scans apparently revealing a tear in his right elbow – although there has been no official comment.
Any significant damage to that region, a common threat to tennis players, would rule him out of the clash in Rouen next week, with his brother urging him to rest.
"It's some sort of tear in his elbow, so he needs to rest," said Jamie Murray.
"He said he can do everything except serve and he told me that rest was all he had to do. I am not planning that he is going to be there (against France) but, if he is, then obviously that is great for the team and we'll see what happens.
"But the most important thing for him is just to get healthy because he has had a few issues now. He has had shingles, he has had his elbow, he was sick here as well for two or three days after he pulled out of the tournament.
"I think he just needs to get a bit settled and get a good crack at it over the next three months because there are a lot of big tournaments to play."
Meanwhile, the Telegraph have suggested the 29-year-old could be sidelined until the clay court season, with a return unlikely before the Madrid Masters in May.
Murray is enduring a topsy-turvy season, with a title in Dubai tempered by early exits in the Australian Open and Indian Wells.
Read the original article on Eurosport: Andy Murray set to miss Davis Cup quarter-final with elbow ‘tear’