Given a random walk (S_n) with typical step distributed according to some fixed law and a fixed parameter p in (0,1), the associated positively step-reinforced random walk is a discrete-time process which performs at each step, with probability 1-p, the same step as (S_n) while with probability p, it repeats one of the steps it performed previously chosen uniformly at random. The negatively step-reinforced random walk follows the same dynamics but when a step is repeated its sign is also changed. In this work, we shall prove functional limit theorems for the triplet of a random walk, coupled with its positive and negative reinforced versions when p < 1/2 and when the typical step is centred. The limiting process is Gaussian and admits a simple representation in terms of stochastic integrals, B(t),tp∫0ts-pdBr(s),t-p∫0tspdBc(s)t∈R+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left( B(t) , \\, t^p \\int _0^t s^{-p} \\mathrm {d}B^r(s) , \\, t^{-p} \\int _0^t s^{p} \\mathrm {d}B^c(s) \\right) _{t \\in \\mathbb {R}^+} \\end{aligned}$$\\end{document}for properly correlated Brownian motions B, B^r, B^c. The processes in the second and third coordinate are called the noise reinforced Brownian motion (as named in [1]), and the noise counterbalanced Brownian motion of B. Different couplings are also considered, allowing us in some cases to drop the centredness hypothesis and to completely identify for all regimes p in (0,1) the limiting behaviour of step reinforced random walks. Our method exhausts a martingale approach in conjunction with the martingale functional CLT.