Curry, who has won 45 caps and skippered England for the first time in last season’s Six Nations opener against Scotland, was hurt during Sale’s Gallagher Premiership victory over Harlequins on Sunday.
England head coach Borthwick is already without Exeter hooker Luke Cowan-Dickie for at least the start of the tournament after he suffered an ankle injury last weekend.
England kick off their campaign at home to Scotland on February 4 – Borthwick’s first game in charge – before hosting Italy eight days later.
Curry has excelled on the Test match stage since making his England debut in 2017, and he is one of the most efficient back-row operators in world rugby.
Sale rugby director Alex Sanderson said: “This is a (grade) 2c hamstring tear, so it wasn’t as bad as originally thought. We got him scanned, assessed and reassessed.
“A 3c would have been six-eight weeks (out), and he would have been looking to play at the back end of the Six Nations.
“With it being a 2c and Tom being very diligent on his rehab, it means he could be back for Wales, certainly Wales (on February 25).
“So that is a real positive for England and all of us, because I want England to do well.
“He gets on with things quick, Tom. He has got his head around it, and he is focused on the Six Nations.
“Now that he knows it could have been six-eight (weeks), but more likely four-six, he is more positive about it.”