The two 23-year-olds feature in the 13-man squad which will take on New Zealand at Lords next month.
The squad was selected by a revamped panel which is led by the new director of men's cricket Rob Key, and includes captain Ben Stokes as well as new head coach McCullum.
Brook, who represents Yorkshire, made his Twenty20 debut last January and has enjoyed a solid start in the LV=Insurance County Championship.
The batsman has already raced to the top of Division One's scoring charts, having produced three centuries and an average of 151.60.
Potts, on the other hand, has caught the eye for Durham in Division Two. The player has produced 35 wickets at 18.57 and has clearly impressed the selectors.
The pair will be joined by the returning James Anderson and Stuart Broad, who were controversially dropped for the recent tour of West Indies.
Key, who also involved performance director Mo Bobat, head scout James Taylor and player identification lead David Court in selection, said: "This is the start of a new era for our Test team under the stewardship of Ben and Brendon.
"With a blend of youth and experience, we have selected an exciting squad that can compete with New Zealand in next month's Test series.
"We have rewarded players in Harry Brook and Matty Potts who have had outstanding starts to the county season, and they deserve the opportunity to stake a claim at this level.
"It promises to be a mouth-watering series, and I can't wait for the team to start against a very good New Zealand side. It is a fascinating prospect for everyone connected with the sport in this country."
England Men's Test squad vs New Zealand:
Ben Stokes (Durham) captain
James Anderson (Lancashire)
Jonathan Bairstow (Yorkshire)
Stuart Broad (Nottinghamshire)
Harry Brook (Yorkshire)
Zak Crawley (Kent)
Ben Foakes (Surrey)
Jack Leach (Somerset)
Alex Lees (Durham)
Craig Overton (Somerset)
Matthew Potts (Durham)
Ollie Pope (Surrey)
Joe Root (Yorkshire)