Hamilton will earn $34.5 million US over the course of the deal, the reports said. That translates to an annual salary-cap hit of $5.75 million.
The 22-year-old was due to become a restricted free agent on Wednesday, when the NHL's signing period opens.
Hamilton was acquired by the Flames last week in a trade with the Bruins. Calgary sent a first-round pick and a pair of second rounders, all in this year's draft, to Boston, which had lost faith in its ability to re-sign Hamilton.
The promising youngster is expected to bolster a Flames blue-line that already includes Mark Giordano, Kris Russell, T.J. Brodie, Dennis Wideman, Ladislav Smid and Deryk Engelland.
Hamilton had 10 goals and 32 assists in 72 games last season for Boston.