No. An arcane focus is made specifically to be an arcane focus. DMs can houserule this differently, but by default an item must be explicitly constructed to be an arcane focus (PHB, p. 151; emphasis mine):
An arcane focus is a special item — an orb, a crystal, a rod, a specially constructed staff, a wand-like length of wood, or some similar item — designed to channel the power of arcane spells.
An object just happening to have the form of an orb, crystal, rod, staff, wand, or "some othersimilar item" (which covers a lot!) isn't enough for it to be used as an arcane focus. Finding a crystal in a cave, a shepherd's rod by the side of a field, a child's glass marble, or a little jade statue of a wizard doesn't mean you've found your new arcane focus.
Similarly, although magic items might happen to take one of these forms, they are not specially constructed to be arcane foci. Yes, they're specially constructed — but they're constructed to be a specific magic item, which has a different magical function from an arcane focus designed to channel outside spells.
(Someone might argue that magic items are already designed to channel some kind of magic, so why not outside spells. To this I answer that arcane magic is arcane and detailed in the particulars of how it works, not wishy-washy — the construction matters very much for what kind of magical use it later has and isn't a general purpose "vessel" for any kind of magic at all. Besides, if being merely enchanted at all makes a wand of fireballs usable as an arcane focus wand, then the broadness of "some similar item" would feasibly make a stone of controlling earth elementals or a wind fan usable as arcane foci.)
Just like normal items, magic items have to explicitly say they may be used as an arcane focus in order to be used as one (unless, like a bard, you have a class feature that says otherwise). An example is the Tome of the Stilled Tongue (DMG, p. 208), which says that it can be used as an arcane focus once attuned. This explicit exception proves the rule that magic items cannot be used as arcane foci unless they are made to be.