He was arrested last April on suspicion of phone hacking and remains on police bail.
Writing in his Huffington Post blog, Mr Thurlbeck hailed Murdoch's decision to begin printing this Sunday, suggesting that the move had "left the opposition in the starting blocks".
But he warned: "By turning the Sun into a seven day operation, Murdoch must avoid doing this on the cheap and thinking he can get away by simply using Sun staff on a seven day rota."
Mr Thurlbeck said there needed to be a "pool of talented Sunday tabloid journalists" who require a "totally different skill set" from daily reporters, adding: "He needs to look no further than his reporters on the News of the World.
"It takes years to discover and then train up a Sunday reporter and every single one of my former colleagues would be a significant asset to the Sun on Sunday.
"I take this opportunity to remind Mr Murdoch that his former CEO Rebekah Brooks promised to find as many News of the World staff as possible new jobs.
"He now has the opportunity to carry out that company promise and has a moral duty to do so.
"I suspect high on his hit list will be the News of the World's former investigations editor Mazher Mahmood who would be back on natural territory. At this point, it must be said that I exclude myself from this process, for dozens of reasons!"