There are several notions of intuitionistic continuum, the closest ones to Aristotle's are Brouwer's "fluid continuum", and especially late Weyl’s version of it since On the New Foundational Crisis of Mathematics (1921). We have to keep in mind, however, that Brouwer and Weyl received their view through a major intermediary, Kant. Although Aristotle’s and Kant’s descriptions of motion and continuum are sometimes indistinguishable phenomenologically, what was objective reality to Aristotle was only a phenomenal form of perception to Kant. But all of them did share the most basic premise: continuum is given as a whole, points and parts are imposed on it.
Weyl makes suggestions as to mathematical realization of his fluid continuum and Brouwer built a full blown theory of his somewhat "less" fluid one, but remarks pessimistically that “the inapplicability of the simple laws of classical logic eventually results in an almost unbearable awkwardness”, the closer we get to the intuitive continuum the less palatable it becomes mathematically. Predicative continuum of early Weyl, developed by Feferman and applied to physics by Field, is even less fluid, but was shown to be sufficient for all of classical physics at least. In his time Aristotle could afford to be optimistic, for one finds the same conception of “fluid magnitude” in his Physics, as in Euclid’s Elements. In our time intuitionism could only build chain of continua mediating between philosophical insight and mathematical physics.
Weyl concludes that there is a divide between mathematical theorizing and philosophical insight into our experience, of time and motion in particular, which seems to echo Aristotle’s “response to the question” vs. “response to the actual facts of the matter”. But he goes further suggesting that it can not be bridged, which implies that Zeno’s challenge must get different answers on different grounds:”if phenomenal insight is referred to as knowledge, then the theoretical one is based on belief... But where is that transcendent world carried by belief, at which its symbols are directed? I do not find it, unless I completely fuse mathematics with physics and assume that the mathematical concepts of number, function, etc. (or Hilbert’s symbols), generally partake in the theoretical construction of reality in the same way as the concepts of energy, gravitation, electron, etc.”
Tieszen gives a nice review of Weyl’s fluid continuum in Philosophical Background of Weyl's Mathematical Constructivism, also his with co-authors Brouwer and Weyl: The Phenomenology and Mathematics of the Intuitive Continuum compares it to Brouwer’s.
P.S. In On the New Foundational Crisis of Mathematics (1921) Weyl wrote: “an ensemble of individual points is, so to speak, picked out from the fluid paste of the continuum. The continuum is broken up into isolated elements, and the flowing-into-each-other of its parts is replaced by certain conceptual relations between these elements, based on the “larger-smaller” relationship. This is why I speak of an atomistic conception of the continuum”. Weyl speaks here of points “picked out” arithmetically, as in the classical conception or his earlier constructivist one.
To approach fluid continuum we need to commit Brouwer’s “second act of intuitionism” and introduce lawless choice sequences that reflect the fluidity of intuitive continuum. In Philosophy of Mathematics and Natural Science (1949) Weyl explains: “the notion of sequence changes its meaning: it no longer signifies a sequence determined by some law or other, but rather one that is created step by step by free acts of choice, and thus remains in statu nascendi. This ‘becoming’ selective sequence represents the continuum, or the variable, while the sequence determined ad infinitum by a law represents the individual real number falling into the continuum. The continuum no longer appears, to use Leibniz’s language, as an aggregate of fixed elements but as a medium of free ‘becoming’”. This is how intuitionistic points only exist "potentially", in Weyl's view unlike Brouwer's, points in the lawless part of the continuum can not even be individuated.