The analysis of expensive numerical simulators usually requires metamodelling techniques, among which Gaussian process regression is one of the most popular approaches. Frequently, the code outputs correspond to physical quantities with a behavior which is known a priori: Chemical concentrations lie between 0 and 1, the output is increasing with respect to some parameter, etc. In this paper, a framework for incorporating any type of linear constraints in Gaussian process modeling is introduced, this including common bound and monotonicity constraints. The proposed methodology mainly relies on conditional expectations of the truncated multinormal distribution and a discretization of the input space. When dealing with high-dimensional functions, the discrete-location approximation requires many points and subsequent integral approximations suffers from the curse of dimensionality. We thus developed a sequential sampling strategy where the input space is explored via a criterion which maximizes the probability of respecting the given constraints. To further reduce the computational burden, we also recommended a correlation-free approximation to be used during the sequential sampling strategy. The proposed approaches are evaluated and compared on several analytical functions with up to 20 input variables, with bound or monotonicity constraints.