Rumours about the state of the pair's marriage emerged last week when Moore posted several cryptic messages on Twitter.
Moore tweeted a quote from Greek philosopher Epictetus on September 23: "When we are offended at any man's fault, turn to yourself and study your own failings. Then you will forget your anger."
Kutcher appeared to dismiss the split rumours by posting a link to the Public Enemy song 'Don't Believe the Hype' on his Spotify account.
On Thursday, photographs were published of a June meeting between Kutcher and a woman bearing a "remarkable resemblance" to Sara Leal, who is apparently asking for $250,000 (£160,000) to keep details of their relationship private.
Kutcher returned to Twitter on Thursday night to seemingly address the continuing rumours.
He wrote: "When you ASSUME to know that which you know nothing of you make an ASS out of U and ME (sic)"
Kutcher and Moore began dating in 2003 and married two years later in a ceremony officiated by a Kabbalist rabbi.
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