The 27-year-old was substituted after just 15 minutes in the Hammers' 3-1 win at Bournemouth.
Carroll had scored two goals in West Ham's past four games after returning to action in September 2015, following seven months out for knee surgery.
"He's a positive lad, but of course he is gutted for several reasons," said Hammers boss Slaven Bilic.
"He felt he was back and enjoying his football.
"It's not the end of the world, though. If everything goes well he will be back very soon so there is no reason to be negative.
"If it goes as we expect, then he will be back for a significant part of the season, the last 10-15 games, with enough time to make a huge impact on our season."
A former Newcastle player, Carroll joined West Ham from Liverpool for a club record £15m in June 2013.
Carroll has won nine England caps and scored twice but has not played for the national team since a World Cup qualifier against San Marino in October 2012.
West Ham are fifth in the Premier League, just a point behind fourth-placed Tottenham with 17 games remaining.

Two armed men wearing masks broke into the 44-year-old victim's house in Meadowbank Terrace at about 00:50.
During the assault the men knocked the victim to the ground and hit him repeatedly with a blunt object before fleeing the property.
Police Scotland have appealed for witnesses.
The first attacker was white with a pale complexion, in his early 20s, 5ft 10ins tall and had dark hair growth on his face, brown eyes and dark unkempt hair.
He was of skinny build and was wearing a light grey cotton zip-up tracksuit with the hood up, grey gloves, white Reebok classics-style trainers with air holes and a grey scarf.
The second attacker is in his early to mid 20s, between 5ft 6ins and 5ft 8ins tall and of skinny build.
He spoke with a local accent. He was wearing an old, black, waist-length outdoor jacket with a dark scarf covering his face and his hood up. He was wearing dark gloves.
Det Insp Alan O' Brien, of Police Scotland, said "This is a particularly vicious attack on a disabled man in his own home and we would ask anyone who saw or heard anything suspicious in the Meadowbank Terrace area in the early hours of Monday morning is asked to contact police immediately.
"In addition, anyone with any information that can help us with our inquiries and assist in tracing the men responsible should also get in touch."

Bell travelled around the Middle East in the early 1890s and is credited with drawing up the borders of modern Iraq.
Her Grade II* listed childhood home, Red Barns, in Redcar, has since been a pub and hotel and is now empty.
Redcar Labour MP Anna Turley is seeking help to save the building and use it as a museum, learning centre and tourist attraction.
"Red Barns has huge significance to the local area and it is tragic that such a beautiful building with so much history has deteriorated into such a poor state," she said.
The current owners have planning permission to convert the building into flats, but are "sympathetic to the building's relevance to the community", she said.
The manager of Bell's photographic archive at Newcastle University, Dr Mark Jackson, said it was "desperately sad" her home had fallen into disrepair.
The house was designed by renowned Arts and Crafts movement architect, Philip Webb, and the interiors were by designer and artist William Morris "so it's an architecturally important building", he added.
Bell left a "fantastic record" of buildings and communities long since destroyed, including 50 photographs of Palmyra in 1900, much of which has been destroyed by so-called Islamic State, Dr Jackson said.
In 2015 Nicole Kidman starred in a film about Bell's life, Queen of the Desert.
West Ham striker Andy Carroll is facing up to six weeks out of action with a hamstring injury.
[NEXT_CONCEPT]
A disabled man has been left with serious injuries after being attacked in his own home in Edinburgh.
[NEXT_CONCEPT]
A campaign has been launched to save the childhood home of the explorer, writer and archaeologist Gertrude Bell.