Harvey, 21, scored five goals in 50 league appearances for the Pilgrims but rejected the offer a new contract.
Manager Gary Mills is also considering signing ex-Burton Albion striker Shaun Harrad, who has been training with the Dragons.
Harrad is a free agent after finishing the 2015-16 season at Torquay, who are keen to re-sign the 31-year-old.
"He's experienced, knows the leagues and has scored an abundance of goals in his career," Mills told BBC Wales Sport.
"I managed him at Notts County when he was a younger player so he could well be part of us as well."
Full-back Jordan Evans has also been on trial with the National League side following his release from Championship club Fulham.
Evans is a product of Wrexham's academy and Mills said the 20-year-old could sign for the club ahead of their opening league game against Dover Athletic on 6 August.
Wrexham have signed Plymouth Argyle forward Tyler Harvey on a one-year deal.