The Scotland international has been a regular under the Canaries boss since he replaced Daniel Farke in November but was not involved in Sunday's FA Cup success over Charlton.
Gilmour, who was singled out for criticism from a small section of the Norwich faithful at Crystal Palace last month, sustained an injury in training and is set to miss the club's remaining games in January.
The expected recovery period will be welcome news for Scotland head coach Steve Clarke, who will be hoping to have the 20-year-old fit and ready for the World Cup play-offs, which start with the semi-final against Ukraine at Hampden on March 24.
Smith, speaking ahead of Wednesday's trip to West Ham, said: "Billy is out for three to four weeks and has gone back to Chelsea for them to assess it.
"Hopefully he will be back for treatment and start rehab with us over the next few days."
There has been some suggestion that Chelsea may recall Gilmour in the wake of the recent criticism, with manager Thomas Tuchel expected to make a decision before the end of the January window.
Smith, however, does not anticipate Gilmour's loan being ended early, adding: "No, I don't believe so. I don't think the injury is significant enough for that.
"I believe it is an injury where he will be assessed by Chelsea and then sent back to us. Hopefully in the next three weeks he is back playing."