The star centre picked up the problem while on Leicester Tigers duty against the Ospreys.
England were initially confident that the power-packed back would be available for the Calcutta Cup clash but he has still not returned to training an is instead targeting the round two encounter with Ireland.
"By the end of the week we expect Manu to be running fully and changing direction," Lancaster said.
"He will be training fully by Tuesday or Wednesday next week, but I'll need to make a call earlier in the week. He won't be training properly until at least Friday.
"Manu is a big loss - he was outstanding against New Zealand. But he missed the first two games of last year's Six Nations because of a hamstring and it's an opportunity for someone else."
The loss of Tuilagi, who ran riot in England's famous victory over New Zealand, is a major blow to Lancaster on the eve of the Championship.
Saracens star Brad Barritt is expected to remain at centre-stage and is likely to form a new midfield partnership with Billy Twelvetrees or Jonathan Joseph, while fly-halves Toby Flood and Owen Farrell are also options.
"We have a pretty good idea of how we'll rejig the midfield. We have our thoughts," said Lancaster.
"Brad will be in the equation and we need to make sure we can still break the gain line and have physical defenders. We've got to have the ability to get across the gain line and defend."