Sources close to Lohan tell us ... Lohan has NOT hired Alvarez on a permanent basis ... and it's unclear how the two hooked up in the first place.
Alvarez reportedly told LAPD detectives that Dr. Conrad Murray had asked him to hide bottles of Propofol in MJ's house on the day the singer died.
As TMZ previously reported, Alvarez was fired from his gig with the Jackson family last year.