This study aimed to compare the concordance between CP and MLSS estimated by various models and criteria and their agreement with MMSS. After a ramp test, 10 recreationally active males performed four to five severe-intensity constant-power output (PO) trials to estimate CP and three to four constant-PO trials to determine MLSS and identify MMSS. CP was computed using the three-parameter hyperbolic (CP3-hyp), two-parameter hyperbolic (CP2-hyp), linear (CPlin), and inverse of time (CP1/Tlim) models. In addition, the model with the lowest combined parameter error identified the "best-fit" CP (CPbest-fit). MLSS was determined as an increase in blood lactate concentration ≤1 mM during constant-PO cycling from the 5th (MLSS5-30), 10th (MLSS10-30), 15th (MLSS15-30), 20th (MLSS20-30), or 25th (MLSS25-30) to 30th minute. MMSS was identified as the greatest PO associated with the highest submaximal steady-state V˙O2 (MV˙O2ss). Concordance between the various CP and MLSS estimates was greatest when MLSS was identified as MLSS15-30, MLSS20-30, and MLSS25-30. The PO at MV˙O2ss was 243 ± 43 W. Of the various CP models and MLSS criteria, CP2-hyp (244 ± 46 W) and CPlin (248 ± 46 W) and MLSS15-30 and MLSS20-30 (both 245 ± 46 W), respectively, displayed, on average, the greatest agreement with MV˙O2ss. Nevertheless, all CP models and MLSS criteria demonstrated some degree of inaccuracies with respect to MV˙O2ss. Differences between CP and MLSS can be reconciled with optimal methods of determination. When estimating MMSS, from CP the error margin of the model estimate should be considered. For MLSS, MLSS15-30 and MLSS20-30 demonstrated the highest degree of accuracy.