The South Africa captain was with the Toffees for several seasons before a move to White Hart Lane a year ago, where injuries and Luka Modric stood in the way of his first team ambitions.
He then went on loan to the Toffees, where he blossomed as a leader in the midfield and scored a vital equaliser against Manchester United to keep the title race going.
Moyes said after the 4-0 win against Fulham on the weekend: "I don't know if we have, or haven't got a chance of keeping Steven Pienaar yet.
"We'll try after the season finishes. I'm sure we would all like to keep him, and I actually think Steven would like to stay.
"If that's the case it gives us a chance to make it happen, obviously he's owned by Tottenham Hotspur but he's told me he wants to stay.
"You can see how terrific he was against Fulham. He created two goals and all around he looks as if he enjoys his football and feels appreciated. What he does is exceptional."