In this paper, we introduce and study fused lasso nearly-isotonic signal approximation, which is a combination of fused lasso and generalized nearly-isotonic regression. We show how these three estimators relate to each other and derive solution to a general problem. Our estimator is computationally feasible and provides a trade-off between monotonicity, block sparsity, and goodness-of-fit. Next, we prove that fusion and near-isotonisation in a one-dimensional case can be applied interchangably, and this step-wise procedure gives the solution to the original optimization problem. This property of the estimator is very important, because it provides a direct way to construct a path solution when one of the penalization parameters is fixed. Also, we derive an unbiased estimator of degrees of freedom of the estimator.