It seems all the answers so far approaching this from a theoretical perspective are approaching this in terms of exact answers, but we can say a lot about when good approximations are possible too. Of course, some answers have already provided silly ways to do this exactly, so approximations may seem unnecessary, but it provides a nice avenue for some basic transcendental number theory.

It is an unsolved problem, which virtually everyone believes to be true, that $\frac e \pi$ is irrational. Let's assume for the moment that this is true. Then it's a trivial corollary of a well-known theorem that if $\alpha$ is an irrational number, and $\beta$ is any real number, there exist arbitrarily good approximations $p + q \alpha \approx \beta$ with $p,q$ integers. That means, taking $\alpha = \frac e \pi$ and $\beta = \frac \phi \pi$, we can find integers $p,q$ such that $p e + q \pi$ approximates $\phi$ to any tolerance you desire.

One such approximation could be $357 \pi - 412 e = 1.61646... \approx 1.61803... = \phi$, which is accurate to one part in 1000. One can do better, but this at least demonstrates the principle. If the 357 and 412 bother you, you may imagine that I've written a sum with 729 terms on the left hand side instead, 357 of which are $\pi$ and 412 of which are $-e$.

So what if, against all bets, $\frac e \pi$ is rational? Then the opposite is true. There is a single best approximation to $\phi$ of the form $p e + q \pi$, which is not exact, and there are infinitely many choices of $p$ and $q$ which yield the same approximation. This is because, in that case every number of the form $p e + q \pi$ is a rational multiple of $e$ with denominator dividing $d$ the denominator of $\frac e \pi$ when written as an integer fraction in lowest terms. Of course, none of these can be exact, since they're all either 0 or transcendental, while $\phi$ is algebraic, and since the set of all such numbers is discrete (being just $\frac{e}{d}\mathbb Z$ where $d$ is the denominator mentioned above), $\phi$ is not in its closure. That is to say, the irrationality of $\frac e \pi$ is equivalent to the existence of arbitrarily good approximations to $\phi$ of the form $p e + q \pi$ for integers $p$ and $q$. Of course, the current lower bounds on $d$ are likely to be extremely large since we know plenty of digits of both $e$ and $\pi$ and haven't yet found any such rational number with value $\frac e \pi$, so there are going to be very good approximations for all practical purposes, but eventually there has to be a single best one, in exactly the same way that there's a single best integer approximation to $\phi$ (namely 2).

Luckily, even in this case we can still construct arbitrarily good approximations to $\phi$ based on $e$ and $\pi$; just not in the same way. Of course, for some $n$, it must be true that $\sqrt[n] \frac{e}{\pi}$ is irrational (this is true for any real number other than 0 and 1, and $\frac e \pi$ is clearly neither). We can play exactly the same game as we did before to get arbitrarily good approximations of the form $p \sqrt[n] e + q \sqrt[n] \pi$ to $\phi$ with $p$ and $q$ integers. If the appearance of this $n$ bothers you, we can even take $n$ to be a power of 2 so that $\sqrt[n] {}$ can be written as a repeated composition of $\sqrt {}$, i.e. $\sqrt[8]{x}=\sqrt {\sqrt {\sqrt{x}}}$.

Note that in all cases above, it's (as far as I know) unknown whether the forms given can exactly represent $\phi$, though all bets are to the negative. Certainly there are no known cases in which it does represent $\phi$ exactly, since that would give a proof that $e$ and $\pi$ are not algebraically independent (a major unsolved problem). In principle, there could be cases where it's definitely known that the form does not represent $\phi$ exactly, but really there's just about nothing about problems like this so it would surprise me if there are any cases known.