"It has been a week since I sustained the injury. I am doing well. It is a long period but I am looking forward to making a return to action," the 28-year-old told the club's website.
"Of course it is a big relief for me that I will not need an operation. I speak from experience and having had a couple of operations it is not an easy thing to cope with.
"It is good news on that front but it doesn't change the time that it will take me to recover."
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Payet's injury means Michail Antonio could get the nod to make just his second Premier League appearance of the season when the sixth-placed Hammers take on derby rivals Tottenham Hotspur on Sunday.
Spurs are a place above West Ham on goal difference and while Antonio admits the injuries to Payet and Enner Valencia are a blow, he is relishing the chance to break into the team.
"Definitely, Dimitri getting injured is a great loss as well as Enner... (but) their absence gives other players the chance to become to star of the club and allow them to show what they can do," the English midfielder said.
"Dimitri has been absolutely superb so far so he will be greatly missed. It's time for the other boys to step up," he added.
"When I joined, I knew it wasn't going to be easy to break through and stake my claim in the Premier League. It's one of those things where I need to wait and bide my time."
(Reporting by Simon Jennings in Bengaluru; Editing by John O'Brien)
