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.

The Giants got a short-handed opener through player-coach Derrick Walser but Jeff Dimmen pulled the Panthers level.
In the second period Mike Forney's powerplay breakaway goal make it 2-1 with Andy Bohmback equalising.
Mike Radja scored from an acute angle and James Desmarais made it 4-2 before Nottingham's David Ling got one back.
The teams meet again on Saturday night in the second match of their double-header.
Nottingham lead the league on 28 points, with Sheffield Steelers on 27 and Belfast 25.
Wrexham have signed Plymouth Argyle forward Tyler Harvey on a one-year deal.
[NEXT_CONCEPT]
The Belfast Giants shaded a 4-3 win over Nottingham Panthers on Friday night to close the gap on the Elite League leaders to three points.