The 26-year-old Vernon, who becomes Mark McGhee's fifth summer signing, was with Colchester United last season.
The Dons have already added Rory McArdle, Josh McGennis and Mark Howard, with McGhee keen to further bolster his squad before the start of the season.
Folly began his career at St Etienne before spending spells at Southampton, Sheffield Wednesday and Plymouth.
He also enjoyed a loan spell at Dagenham & Redbridge last season.
Vernon began his career at Oldham Athletic and counts Blackpool and Northampton among his former clubs.
Aberdeen are waiting to hear whether midfielder Lee Hendrie, who is currently without a club after leaving Derby County, will agree terms at Pittodrie.
MY SPORT: DEBATE
Give your reaction to this story
Defender Robbie Williams, who was not offered a new contract by Huddersfield Town at the end of last season, may also be offered a deal.
And the Dons expect to sign former Hearts and Celtic midfielder Paul Hartley, who is expected to leave Bristol City.
Hearts have ruled out a move for the 33-year-old.
Source: BBC Sport
Source: BBC Sport