The 26-year-old, however, remains on the sidelines due to some inflammation around the ankle, pushing his likely return to next month.
"Jack felt some pain the last two days here so he needs more time to recover," Pellegrini told the British media.
The Chilean manager was more optimistic about Andy Carroll's fitness levels, with the striker penned for a return in "15 days" after suffering an ankle issue during pre-season.
West Ham, currently 15th in the Premier League standings with seven points, host London rivals Tottenham Hotspur on Saturday.
(Reporting by Hardik Vyas in Bengaluru)
