You can always apply the algorithms to any data sets. You can also always pre-whiten (or drop the columns that cause collinearity) from
your data matrix. The question, IMO, is therefore not whether these algorithms can be computed on (nearly) collinear data matrices $\pmb X$ but whether the estimates you obtain from these clustering algorithms are affected by the covariance structure of the data.
(The argument below can be slightly modified to also account for soft clustering algorithms, but that would make the explanation more complex because it will require the introduction of a weight function, so for simplicity's sake I will stick to hard clustering algorithms.)
Given $\pmb X$ an $n$ by $p$ dataset with rows $\{\pmb x_i\}_{i=1}^n$, a hard clustering algorithm realizes a partition of $\{1:n\}$ into $J>1$ non overlapping subsets $\{C_j\}_{j=1}^J$.
Consider the results of such a partition. The argument below could be made of other estimates derived from the partitions $\{C_j\}_{j=1}^J$ (see paper linked below) but for simplicity's sake I will keep the focus on the estimated cluster centres $\{\pmb t(\pmb X,C_j)\}_{j=1}^J$: each $\pmb t(\pmb X,C_j)$ is a $p$-vector and gives a location estimate for the corresponding cluster. For example, $\pmb t(\pmb X,C_j)$ could be the intra cluster mean:
$$\pmb t(\pmb X,C_j)=\mbox{ave}_{i\in C_j} \pmb x_i$$
A location estimator $\pmb t(\pmb X,C_j)$ for which it holds that:
$$(1)\quad \pmb t(\pmb A\pmb X,C_j)=\pmb A \pmb t(\pmb X,C_j)$$
for any non singular $p$ by $p$ matrix $\pmb A$ is said to be affine equivariant.
From a statistical point of view, affine equivariant location estimates are not affected by the covariance structure of the data matrix $\pmb X$. This is because when a non singular transformation is applied to the data, these estimates transforms like the data. Therefore, they yield statistically equivalent estimates whether they are ran on $\pmb X$ or on a whitened copy of $\pmb X$. This definition can be extended to also hold even if some columns of $\pmb X$ are nearly collinear (a related but stronger property called exact fit insures that the location estimates also transform like the data when $\pmb X$ contains exactly collinear columns or when $\pmb A$ is singular).
My understanding is that the OP is essentially asking (at the minimum) for clustering methods that yield affine equivariant estimates (from a statistical point of view, such estimates would not be affected by the covariance structure of the data). At the maximum, the OP is asking for clustering methods that yield estimates having the exact fit property.
A well known result [0] (ungated copy) is that $(1)$ (and the exact fit property) holds only if $\#\{C_j\}/n>1/2$. This condition excludes all clustering methods as defined in the intro to this answer.
- (0) H. P. Lopuhaa and P. J. Rousseeuw (1991). Breakdown Points of Affine Equivariant Estimators of Multivariate Location and Covariance Matrices. Ann. Statist. Volume 19, Number 1, 229-248.