Motivated by a number of theoretical considerations, predicting the deeply bound double-heavy tetraquarks T[u¯d¯]{bb}, T[u¯s¯]{bb} and T[d¯s¯]{bb}, we explore the potential of their discovery at Tera-Z factories. Using the process Z→bb¯bb¯, we calculate, employing the Monte Carlo generators MadGraph5_aMC@NLO and Pythia6, the phase space configuration in which the bb pair is likely to fragment as a diquark. In a jet-cone, defined by an invariant mass interval mbb<MT[q¯q¯′]{bb}+ΔM, the sought-after tetraquarks T[q¯q¯′]{bb} as well as the double-bottom baryons, Ξbb0,−, and Ωbb−, can be produced. Using the heavy quark–diquark symmetry, we estimate B(Z→T[u¯d¯]{bb}+b¯b¯)=(1.2−0.3+1.0)×10−6, and about a half of this for the T[u¯s¯]{bb} and T[d¯s¯]{bb}. We also present an estimate of their lifetimes using the heavy quark expansion, yielding τ(T[q¯q¯′]{bb})≃800 fs. Measuring the tetraquark masses would require decays, such as T[u¯d¯]{bb}−→B−D−π+, T[u¯d¯]{bb}−→J/ψK‾0B−, T[u¯d¯]{bb}−→J/ψK−B‾0, T[u¯s¯]{bb}−→Ξbc0Σ−, and T[d¯s¯]{bb}0→Ξbc0Σ¯0, with subsequent decay chains in exclusive non-leptonic final states. We estimate a couple of the decay widths and find that the product branching ratios do not exceed 10−5. Hence, a good fraction of these modes will be required for a discovery of T[q¯q¯′]{bb} at a Tera-Z factory.