The tweet came with several anti-whaling campaigns hashtags.
Chief Cabinet Secretary Yoshihide Suga said the site was down but would not confirm the Anonymous claim.
A person claiming association with Anonymous had tweeted early on Thursday morning to Mr Abe, saying "Whaling IS NOT Cultural Right! Your website is #TangoDown!". The identity of the person could not be verified.
Japan resumed its whaling operation in the Antarctic this month after a year`s hiatus, despite international opposition. It says it is conducting scientific research - and that whales meat which is sold to consumers is a by-product of programme.
Eating whale meat, while not common, is considered a cultural tradition by some Japanese. Critics say the whaling programme is inhumane and unsustainable.
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