The Netherlands international netted a late winner in the 2-1 fourth-round triumph at Bolton on Saturday to keep Everton's cup hopes alive.
Heitinga has been linked with a move away from Goodison Park during this month's transfer window after finding regular playing time limited this campaign, taking a back seat to Phil Jagielka and Sylvain Distin.
But it is thought Moyes is keen to keep his squad intact and 29-year-old Heitinga, voted the club's player of the season last term, is keen to stay and battle for a place.
He told the Liverpool Echo: "I'm working hard to get my place back in the team and I'm a fighter so I'll never give up.
"If the manager makes another decision and doesn't put me in the team that's his decision, and the only thing I can do is focus on the time I do get. Hopefully I can show everybody that I'm still the same person that I was last season.
"The most important thing as a player is that you believe in yourself and give everything you have, then the rest follows.
"The team is performing really well this season, but as a player you need a couple of games to get back at your level. I've spoken to the manager - he's happy with me and I'm happy with Everton."