Contract talks with the Leeds youngster are expected to go smoothly, while his team-mate will also reportedly be offered a new deal soon.
And the Yorkshire side are also readying a contract offer for 19-year-old midfielder Alex Mowatt, report the same paper.
Leeds had reportedly opened contract talks with highly rated youngster Cook around two days ago and are now said to hope to agree new terms with the midfielder by the summer.
Mowatt, meanwhile, could be kept at Elland Road until beyond 2017 should he feel satisfied by the deal being prepared for him.
"Leeds United’s contract talks with Lewis Cook should run their course. The club expect to find common ground without difficulty and hope the 17-year-old will agree new terms by the summer," Phil Hay of the YEP writes.
"In the meantime, there are moves afoot to extend Alex Mowatt’s deal and pin him down beyond 2017."
Mowatt, 19, has been another key part of Leeds' season despite his young age. Interest from the Premier League has in the past been reported, however, with Leeds sporting director Nicola Salerno asserting that the club will hold onto him.
Should Mowatt sign a deal keeping him at Elland Road in the long term, it will certainly help towards that end. And the same, of course, applies for Cook.
Neil Redfearn's Leeds side aren't in action this weekend, having been knocked out of the FA Cup by Sunderland in the third round of the competition. When the Championship campaign resumes next weekend, though, the Yorkshire side will visit Huddersfield Town.