The 23-year-old ruptured his Achilles tendon in training last Thursday, and has returned to his home in Belgium for the operation, according to the Daily Mail.
Club manager Paul Lambert told reporters: "Our doctor (Roddy Macdonald) is over in Belgium and he has spoken with the surgeon, and it's pretty imminent he will get it done - today, tonight or tomorrow.
"Then after that it's all about the rehabilitation with him. When it's a long-term injury you have good days, bad days, like everyone else when they are out injured for a long time, but he'll be fine."
Benteke is unlikely to return to match action until October at the earliest, ruling him out of Belgium's first World Cup campaign since 2002.