Franklin was about 855km southwest of Bermuda with maximum sustained winds nearing 165km/h, the National Hurricane Center (NHC) said on Sunday.
"Additional strengthening is forecast, and Franklin is forecast to become a major hurricane on Monday," the advisory said.
A tropical storm with maximum sustained winds of above 180km/h or higher is considered a major hurricane.
Franklin is expected turn away from the US eastern seaboard in the early part of the week but pass near the island of Bermuda on Wednesday, an NHC map shows.
The NHC said swells generated by Franklin were beginning to affect Bermuda on Sunday night.
"These swells are expected to spread toward the east coast of the United States during the next couple of days, likely resulting in life-threatening surf and rip current conditions," it said in the advisory.
This year's Atlantic hurricane season, which runs from June 1 to November 30, is expected to have 18 named tropical storms, nine of which become hurricanes, four of them major, according to forecasters from Colorado State University.
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