The 21-year-old was a free agent after being released by Ipswich Town at the end of last season.
The left-back was given his chance in the first-team at the Tractor Boys last season by Cook and the pair link-up again.
Clements said: “I’m really excited and looking forward to getting going. Linking up with the gaffer again, it was a no-brainer to come here.
“I’ve heard a lot of good things about the club and the town itself. A couple of loan spells haven’t gone exactly as I would like, but I’m now ready to make an impact.
“[Paul Cook] was a massive factor. He gave me my opportunity at Ipswich, he put his trust in me and I like to think I repaid him.
“I’m hard-working. I get up and down, defensive duties definitely come first but when I can get forward and into the attacking third, I can get a cross in and hopefully get assists.”
Clements had been at Ipswich for 14 years, working his way up through the academy.
In total he made nine appearances for his boyhood club.
He had also had a loan spell at League Two Stevenage last season after Cook departed Portman Road.
Clements becomes Chesterfield’s second left-sided signing this summer, following the arrival of fellow youngster Branden Horton from Doncaster Rovers.