The paper presents a control-oriented modeling method for wet clutch friction, considering thermal dynamics, with a focus on paper-based friction linings. Abrupt engagements of such clutches may lead to discomfort and reduce the overall lifespan. The limitations of feedback control, attributed to restricted sensors and modeling inaccuracies, underscore the effectiveness of model-based control employing precise and invertible models. The study extensively explores the largest model uncertainty in slip control-clutch friction. The proposed model integrates the Coulomb friction coefficient, incorporating variables such as pressing force, friction speed, and the temperature of the friction surface. The proposed torque model enables model inversion, allowing the determination of desired oil pressure from torque requirements. Experimental data, comprising 184 sets, validate the model accuracy, with a focus on temperature effects. The Coulomb friction coefficient model exhibits exponential and linear components, capturing the Stribeck effect and temperature variations. Model verification through experiments demonstrates good agreement, supporting its efficacy for wet clutch control performance. The study contributes insights into wet clutch friction models for model-based control.