In his day, Plato counted as a conservative and his Republic has been regarded as an attempt to give society a rigid, unchanging form. Of course, what counts as conservatism has changed over the years. I don't think most of us would find Sparta or the Republic a good model for our own society. Today, a state run by guardians from on high for the public good, looks very much like modern judicial-bureaucratic liberalism.
But liberals have also been able to build on Aristotle's view of human needs to justify the welfare state. Aristotle represents a common sense path that radicals have found too realistic and elitist. He accepts human nature and institutions too much to win over those who want to change them. But there may be some on the right who find Aristotle, the exponent of the mean between extremes, unappealing.