Campbell, a leg-spin bowling all-rounder, has earned his first call-up to the senior team, having previously represented Zimbabwe’s emerging team in the 13th African Games last month.
He contributed with 115 runs in four innings at a strike rate of 126.37 and took one wicket, helping his team win the gold medal.
His father Alistair Campbell led Zimbabwe in 21 Tests and 86 ODIs from 1996 to 2002.
Meanwhile, top-order batter Tadiwanashe Marumani and allrounder Faraz Akram have earned recalls for the Bangladesh tour under the leadership of Sikandar Raza.
Marumani and Akram missed the T20I series against Sri Lanka, which Zimbabwe lost.
Zimbabwe and Bangladesh will square off in the first of the three-match T20I series in Chattogram on 3 May.
The next two matches will be played at the same venue on 5 and 7 May, while Dhaka will host the last two matches on May 10 and 12.
Zimbabwe T20I squad: Sikandar Raza (c), Faraz Akram, Brian Bennett, Ryan Burl, Johnathan Campbell, Craig Ervine, Joylord Gumbie (wk), Luke Jongwe, Clive Madande (wk), Tadiwanashe Marumani, Wellington Masakadza, Blessing Muzarabani, Ainsley Ndlovu, Richard Ngarava, Sean Williams