  VSe$_2$ is a layered compound that has attracted great attention due to its
proximity to a ferromagnetic state that is quenched by the presence of a charge
density wave (CDW) phase. In the monolayer limit, unrelated experiments have
reported different CDW orders with transition temperatures in the range of 130
to 220 K, making this monolayer very controversial. Here we perform
first-principles non-perturbative anharmonic phonon calculations in monolayer
VSe$_2$ in order to estimate the CDW order and the corresponding transition
temperature. Our analysis solves previous experimental contradictions as we
reveal that monolayer VSe$_2$ develops two independent charge density wave
orders associated to $\sqrt{3} \times \sqrt{7}$ and $4 \times 4$ modulations
that compete as a function of strain. In fact, tiny variations of only 1.5% in
the lattice parameter are enough to stabilize one order or the other, which
underlines that the CDW order becomes substrate-dependent. The predicted CDW
temperature is strain-dependent and has a value of around 220 K, in good
agreement with experiments. Moreover, we analyze the impact of external
Lennard-Jones interactions on the CDW. We show that these can act together with
the anharmonicity to suppress the CDW orders. In the particular case of
monolayer VSe$_2$, this may give rise to the emergence of a ferromagnetic
order.
