The on-loan Liverpool frontman limped off in the second half of his debut against Fulham.
Co-owner David Gold responded to questions on his official Twitter account suggesting the player being back by mid-October would be the "worst-case scenario".
Carroll is now certainly expected to miss the resumption of the domestic season at Norwich next weekend, as well as the Capital One Cup tie against Wigan.
The Irons host Arsenal on October 6, by which time if the striker is not back towards match fitness, he would also be a major doubt for the next round of World Cup qualifiers with San Marino at Wembley and in Poland the following week.
Hammers boss Sam Allardyce is in no doubt Carroll, who played a key role in two of his side's goals in the 3-0 win over Fulham, can be a major asset.
He said: "If Andy scores between ten and 15 goals, it is going to give us a great chance to be out of the relegation zone.
"If Andy gets 15 to 20 goals, we can get in the top ten because all we have to do there is to produce enough clean sheets to make sure we get points on the board with the goals that he scores."