A continuum model based on the contact mechanics theory was developed and used for evaluating virus indentation forces at the early stage of membrane invagination, as well as the work of the virus indentation forces and virus-cell contact pressures in a receptor-mediated endocytosis, depending on the virus size and virus/cell stiffnesses. The model indicated that early virus indentation forces are in the order of 1–10 pN and for a given extent of virus engulfment, they increase linearly with the elastic modulus of the host cell and also with the square of the virus radius. The work of invagination at the initial phase of virus endocytosis is in the order of tens of zeptojoules and peak virus-cell contact pressures at this stage are in the order of hundreds of Pascals to several kPa. For a given extent of virus engulfment, peak and average contact pressures increase linearly with the elastic modulus of the host cell but interestingly, they are negligibly affected by the virus size. The present model may be useful in the fields of cellular biomechanics, virology and nanodrug delivery to evaluate mechanical factors during the early phase of membrane invagination.