Both players have passed medicals at Bootham Crescent and will join up with Nigel Worthington's first-team squad.
Fletcher, 22, who had a spell on loan with Yeovil last season, came through the youth system at Turf Moor. He has also spent time on loan at Grimsby, Stockport, Accrington and Crewe.
Kettings, who made his debut for Scotland Under-21s last year against Canada, has joined the Minstermen on a year-long loan from Blackpool.
Meanwhile, York have appointed Andy Collett as their new goalkeeping coach.
Collett, who has left his post as goalkeeping coach at Hartlepool, made a total of 235 appearances for Middlesbrough, Bristol Rovers and Darlington.