Carbon nanotube (CNT) transistors exemplify the fundamental tradeoff between desirable high mobility and undesirable leakage current due to the small effective mass and bandgap. To understand leakage current limits in high-speed CNT transistors, electrical bandgaps are extracted on 12 single-CNT top-gate MOSFETs from the energy gap between thermionic emission and band-to-band tunneling (BTBT) at 10 K. At 300 K the minimum I<sub>OFF</sub> at 0.5 V V<sub>DS</sub> is analyzed as a function of bandgap between 0.96 eV and 0.43 eV with I<sub>OFF-MIN</sub>from 0.2 pA/CNT to 15 nA/CNT. NEGF simulation validates the bandgap extraction methodology and reproduces the experimental MOSFET I<sub>OFF-MIN</sub> data. A TCAD model calibrated to this work’s leakage data projects the accessible I<sub>ON</sub>-I<sub>OFF</sub> design space bounded by CNT bandgap, indicating E<sub>G</sub> > 0.65 eV (d<sub>CNT</sub> < 1.3 nm) is needed to achieve 100 nA/<inline-formula> <tex-math notation="LaTeX">$\mu \text{m}$ </tex-math></inline-formula> at 0.5 V V<sub>DD</sub> and 250 CNT/<inline-formula> <tex-math notation="LaTeX">$\mu \text{m}$ </tex-math></inline-formula> for channel length above 20 nm. An E<sub>G</sub> of 1.06 eV (d<sub>CNT</sub> = 0.8 nm) can deliver <inline-formula> <tex-math notation="LaTeX">$2750\times $ </tex-math></inline-formula> tunable range of I<sub>OFF</sub> by adjusting V<sub>T</sub>, which exceeds the <inline-formula> <tex-math notation="LaTeX">$400\times $ </tex-math></inline-formula> tunable range of I<sub>OFF</sub> used in Si CMOS platform technologies.