When interval branch and bound solvers are used for solving constrained global optimization, upper bounding the objective function is an important mechanism which helps to reduce globally the search space. Each time a new upper bound UB is found during the search, a constraint related to the objective function f(obj)(x) < UB is added in order to prune non-optimal regions. We quantified experimentally that if we knew a close-to-optimal value in advance (without necessarily knowing the corresponding solution), then the performance of the solver could be significantly improved. Thus, in this work we propose a simple mechanism for estimating upper bounds in order to accelerate the convergence of interval branch and bound solvers. The proposal is validated through a series of experiments.