Summary: Taking account of recent developments in the representation of \(d\)-dimensional isotropic stable Lévy processes as self-similar Markov processes, we consider a number of new ways to condition its path. Suppose that \(\mathsf{S}\) is a region of the unit sphere \(\mathbb{S}^{d-1}=\{x\in\mathbb{R}^d:|x|=1\}\). We construct the aforesaid stable Lévy process conditioned to approach \(\mathsf{S}\) continuously from either inside or outside of the sphere. Additionally, we show that these processes are in duality with the stable process conditioned to remain inside the sphere and absorb continuously at the origin and to remain outside of the sphere, respectively. Our results extend the recent contributions of L. Döring and P. Weissmann [Bernoulli 26, No. 2, 980–1015 (2020; Zbl 1466.60099)], where similar conditioning is considered, albeit in one dimension as well as providing analogues of the same classical results for Brownian motion, cf. J. L. Doob [Bull. Soc. Math. Fr. 85, 431–458 (1957; Zbl 0097.34004)]. As in Döring and Weissman [loc. cit.], we appeal to recent fluctuation identities related to the deep factorisation of stable processes, cf. [the first author, Electron. J. Probab. 21, Paper No. 23, 28 p. (2016; Zbl 1338.60130)], [the first author et al., Potential Anal. 53, No. 4, 1347–1375 (2020; Zbl 1460.60024)] and [the first author et al., Stochastic Processes Appl. 127, No. 4, 1234–1254 (2017; Zbl 1373.60085)].