The South African midfielder enjoyed a successful loan spell at Goodison Park last season having left the club a year ago and the Toffees boss now wants to make the deal permanent
Moyes finalised the capture of free agent Steven Naismith earlier this week following his release from Rangers and hopes the striker will be followed through the Goodison Park entrace by Pienaar.
The South African international returned to the club during the second half of last season on-loan from Spurs, scoring four goals in 14 games as the Toffees narrowly missed out on European qualification.
Having spent three-and-a-half years on Merseyside prior to his switch to White Hart Lane in January 2011, Moyes is hopeful of bringing the 30-year-old back to the club this summer.
"We're keen and I hope in time that we will be able to conclude that deal," he told reporters.
Tottenham appointed Andre Villas-Boas as the successor to Harry Redknapp on Tuesday and Moyes revealed he was unsure as to whether it would curtail his hopes of bringing Pienaar back to Everton.
"I don't know if it has or not," he added.
"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."