The Spaniard came off with the problem early on in Saturday's 2-1 loss at Fulham and will miss the club's festive programme and the majority of January.
On the length of his absence, Bielsa, whose side welcome fifth-placed Preston on Boxing Day, said: "One month. It was an injury in his hamstring.
Bielsa also confirmed that Tyler Roberts and Jamie Shackleton are available for the home clash with North End, although Barry Douglas has been suffering with sickness.
Gjanni Alioski, who came on for Hernandez at Craven Cottage, is likely to start at left-back against Preston, with Stuart Dallas operating in midfield.
Bielsa added: "Tyler and Shackleton are available. Douglas is sick.
"Forshaw, we still don't know when he will come back to the group."
Second-placed Leeds hold an eight-point buffer over Preston, with Bielsa's side trailing leaders West Brom - who they visit on New Year's Day - by three points.