Elsom, 27, picked up the new injury during Australia's end of season tour of Europe.
He has now been ordered to rest in a bid to avoid surgery or a long-term spell on the sidelines.
"I know he's reluctant to have surgery, as are we, because that would put him out for some time," said Brumbies coach Andy Friend.
"It's that far removed as an option at the moment. Our doctor Angus Bathgate has been having discussions with him.
"It's really a time thing, just to see how this little rest period works for him, what that does to his hamstring and whether that will be enough for him."
With Australia and Brumbies legend George Smith currently plying his trade in Toulon, although linked to a move in Japan, the loss of Elsom would be a major blow. But they can turn to experienced back-rower Julian Salvi.
"He adds a lot of experience there, but we've got Colby Faingaa and Michael Hooper, who I think are two quality footballers and it's going to be exciting to see who ends up wearing that jersey," added Friend.
The Lions will tour Australia in 2013 and will be looking to gain revenge for the 2-1 Test series defeat they suffered at the hands of the Wallabies in 2001.