Precise measurements of the branching ratios for the flavor-changing neutral current decays K -> π+νbar+ν can provide unique constraints on CKM unitarity and, potentially, evidence for new physics. The NA62 experiment at the CERN SPS is currently collecting data and will measure BR(K+ -> π++νbar+ν) to within 10%. The yet to be proposed KOTO experiment at JPARC would measure BR(KL -> π0+νbar+ν) with similar precision using a low-energy neutral beam. As a natural continuation of the NA62 physics program, we are designing the KLEVER experiment to measure BR(KL -> π0+νbar+ν) to ~20% using a high-energy neutral beam at the CERN SPS starting in LHC Run 4. Our approach is complementary to that of KOTO. We present findings from feasibility studies, with an emphasis on the design challenges faced and the potential sensitivity for the measurement of BR(KL->π0+νbar+ν).