The centre has issued storm watches for much of the province's coastal areas.
Chris Fogarty of the centre in Halifax said it was hard to predict exactly where the storm will hit because Leslie's circulation is about 800-kilometres in diameter.
Fogarty predicts wind gusts could top 100 kilometres an hour in Newfoundland and Cape Breton.
Newfoundland, Nova Scotia and P.E.I. could also see 100 to 150 millimetres of drenching rain over about a period of about 36 hours.
The storm's outer bands buffeted Bermuda Sunday causing some power outages and littering streets with tree branches and other debris, but there was no major damage or injuries.