Summary: Due to their flexibility, Gaussian processes (GPs) have been widely used in nonparametric function estimation. A prior information about the underlying function is often available. In this article, the finite-dimensional Gaussian approach proposed by Maatouk and Bay in 2017 which can satisfy linear inequality conditions everywhere (e.g., monotonicity, convexity and boundary) is considered. In a variety of real-world problems, the observed data usually possess noise. In this article, this approach has been extended to deal with noisy observations. The mean and the maximum of the posterior distribution are well defined. Additionally, to simulate from the posterior distribution two methods have been used: the exact rejection sampling from the Mode and the Hamiltonian Monte Carlo method which is more efficient in high-dimensional cases. The generalization of the Kimeldorf-Wahba correspondence is proved in noisy observation cases. A comparison shown that the proposed model outperforms all recent models dealing with the same constraints in terms of predictive accuracy and coverage intervals.