Is centering a valid solution for multicollinearity?

Let's assume that $y = a + a_1x_1 + a_2x_2 + a_3x_3 + e$ where $x_1$ and $x_2$ both are indexes both range from $0-10$ where $0$ is the minimum and $10$ is the maximum. I found by applying VIF, CI and eigenvalues methods that $x_1$ and $x_2$ are collinear. Can these indexes be mean centered to solve the problem of multicollinearity?

When the model is additive and linear, centering has nothing to do with collinearity. Centering can only help when there are multiple terms per variable such as square or interaction terms. Even then, centering only helps in a way that doesn't matter to us, because centering does not impact the pooled multiple degree of freedom tests that are most relevant when there are multiple connected variables present in the model. For example, if a model contains $X$ and $X^2$, the most relevant test is the 2 d.f. test of association, which is completely unaffected by centering $X$. The next most relevant test is that of the effect of $X^2$ which again is completely unaffected by centering.

(An easy way to find out is to try it and check for multicollinearity using the same methods you had used to discover the multicollinearity the first time ;-)

No, unfortunately, centering $x_1$ and $x_2$ will not help you. When you have multicollinearity with just two variables, you have a (very strong) pairwise correlation between those two variables. Consider this example in R:

library(MASS)
set.seed(1)
X = mvrnorm(100, mu=c(30,30), Sigma=rbind(c(100,  97),
c( 97, 100) ))
x1 = X[,1]
x2 = X[,2]
cor(x1, x2)
# [1] 0.9698819


Centering is just a linear transformation, so it will not change anything about the shapes of the distributions or the relationship between them. Instead, it just slides them in one direction or the other. To see this, let's try it with our data:

x1c = x1 - mean(x1)
x2c = x2 - mean(x2)
cor(x1c, x2c)
# [1] 0.9698819


The correlation is exactly the same. Here's what the new variables look like:

They look exactly the same too, except that they are now centered on $(0, 0)$.

Now to your question: Does subtracting means from your data "solve collinearity"? One answer has already been given: the collinearity of said variables is not changed by subtracting constants. You can see this by asking yourself: does the covariance between the variables change? Well, since the covariance is defined as $Cov(x_i,x_j) = E[(x_i-E[x_i])(x_j-E[x_j])]$, or their sample analogues if you wish, then you see that adding or subtracting constants don't matter. Hence, centering has no effect on the collinearity of your explanatory variables.