The South African midfielder spent the second half of the previous campaign at Goodison Park scoring four goals in 14 as the Toffees’ narrowly missed out on securing a place in the Europa League.
Pienaar spent three-and-a-half years on Merseyside before moving to Spurs during the 2011 January transfer window and his old boss is hoping to bring him back to the club this summer.
The 30-year-old found first team opportunities hard to come by at White Hart Lane resulting in the temporary switch to his old club with Moyes now eager to make it a permanent deal.
“We’re keen and I hope in time that we will be able to conclude that deal,” said Moyes at a press conference on Thursday.
The North London club appointed Andre Villas-Boas as their new manager replacing Harry Redknapp and Moyes isn’t sure whether that will scupper his chances of re-signing Pienaar.
“I don’t know if it has or not,” he replied.
“I don’t like talking about players at other clubs but I think the fact of the matter is that Steven has made it clear he wants to come back to Everton.
“By my understanding that hasn’t changed. My next job hopefully in the coming weeks is to get Steven Pienaar here as well.”
Moyes is once again working on a shoestring budget at Everton but was able to bolster his attack earlier this week after finalising the capture of Steven Naismith on a free transfer.
The Scotland international terminated is contract at cash-stricken Rangers signing a four-year deal with the Toffees telling their official website it was an easy decision to make.
“To come to such a wonderful Club is extra special,” Naismith said.
“My decision to move here was down to the manager’s leadership, first of all. Over the last 10 years, it’s been one of the most successful clubs in terms of growing.”