The pair released debut album The Age of the Understatement in 2008 and while Kane promised a follow-up, Turner suggested that a new record would not happen in the near future.
Asked if the project is still on hold, Kane told Geoff Lloyd on Absolute Radio that: "Yeah, we've been saying all along it's still on hold and when the time's right we'll both ride that ship again."
Prompted on when a comeback might happen, he added: "Who knows, man... I'll just keep it up my sleeve for now."
Earlier this year, Kane supported Arctic Monkeys at London's Roundhouse venue as part of the iTunes festival.
Kane later joined the band on stage for their V Festival performance in August.
Watch Miles Kane talk about the future of The Last Shadow Puppets below:
Watch The Last Shadow Puppets' 'Standing Next To Me' music video below: