Grealish played the full 90 minutes as Everton beat former club Aston Villa 1-0 in the Premier League on Sunday.
Everton, who are 10th in the table, host Leeds on Monday.
Moyes said they were in “close contact” with City about whether Grealish will stay at Everton during his recovery.
Asked how it impacts the possibility making his transfer permanent in the summer, Moyes said: “We’re not thinking about it at the moment.
“We’re only thinking about Jack. We’re upset and disappointed for him that he’s got an injury.”
Grealish’s form at Everton led to suggestions that he could earn a recall to the England squad and push for a place at the World Cup this summer.
His most recent appearance came under caretaker manager Lee Carsley in October 2024.
Grealish won three Premier League titles, the Champions League and the FA Cup after joining City from Aston Villa for a then British record £100m in 2021.
But he fell out of favour last season, making only seven league starts before being left out of the squad for the Club World Cup in the US.