Milan said in a statement that Biglia had suffered "a severe lumbar trauma" following an injury during Saturday's 1-0 Serie A defeat against Benevento.
"The player has undergone medical examinations that highlighted a fracture of the transverse processes of the first two lumbar vertebrae," said the club in a statement.
"A detailed prognosis will be issued in the coming days upon completion of all medical tests."
According to specialist websites, a fracture of the transverse process is a relatively rare injury that takes four to six weeks to heal but usually requires a further month's recovery time afterwards.
Argentina's first World Cup match is against Iceland on June 16.
Biglia started the quarter-final, semi-final and final of Argentina's campaign in the 2014 World Cup where they finished as runners-up to Germany.
The 32-year-old has been a regular since then and played in most of their 18-match qualifying campaign.
(Writing by Brian Homewood in Berne; Editing by Clare Fallon)
