High-entropy alloys (HEAs) with high melting points and low thermal neutron cross-section are promising new cladding materials for generation III+ and IV power reactors. In this study a recently developed high throughput computational screening tool Alloy Search and Predict (ASAP) has been used to identify the most likely candidate single-phase HEAs with low thermal neutron cross-section, from over a million four-element equimolar combinations. The selected NbTiVZr HEA was further studied by density functional theory (DFT) for moduli and lattice parameter, and by CALPHAD to predict phase formation with temperature. HEAs of NbTiVZrx (x = 0.5, 1, 2) were produced experimentally, with Zr varied as the dominant cross-section modifier. Contrary to previous experimental work, these HEAs were demonstrated to constitute a single-phase HEA system; a result obtained using a faster cooling rate following annealing at 1200 °C. However, the beta (BCC) matrix decomposed following aging at 700 °C, into a combination of nano-scale beta, alpha (HCP) and C15 Laves phases.