Cadmium telluride (CdTe) has prime importance in photovoltaics due to its direct band gap of 1.45 eV. However, its two-dimensional counterparts have not been fully explored due to polymorphism. We investigated the energy phase diagram of 2D CdxTey (x + y <= 8) using state-of-art computational methods and found the phases of CdTe and CdTe2 on and near the energy convex-hull, respectively. Further screening of phonon and ab initio molecular dynamics simulations confirms their experimental viability. These structures reveal promising electronic properties. 2D CdTe has a robust direct band gap unaffected by thickness, monolayer to trilayer and bulk, and strain as high as 7%. Such robust semiconductors are crucial for device applications because of challenges in the growth of wafer-scale uniform monolayers. In contrast, the direct band gap of 2D transition metal dichalcogenides is highly sensitive to thickness and strain, limiting their usage in devices. The 2D CdTe2 has an indirect band gap whose magnitude is tunable by strain. The robust direct band gap of 2D CdTe and the tunable indirect band gap of CdTe2 make them potential candidates for optoelectronic devices.