In fact, Greek and English, being both Indo-European languages, are more closely related than Korean and the various Chinese languages.
There is a controversial theory that Korean, Mongolian, and Machurian, are all a part of an Altaic superfamily — along with Japanese, Turkish, and a number of others. This was broadly accepted fifty years ago, and less so today.
Given that we have writings from Indo-European speaking empires since 1000 BC that we can mine for phonological clues and trace evolutionary changes with while the languages of supposed Altaic only began to be written after 700 AD, it's not surprising that there's less evidence to be found and not enough for a hypothesis bold enough to link those disparate cultures together. The idea that Hindi, Persian, Latin, Russian, and English are linked together is already very surprising.
There is a controversial theory that Korean, Mongolian, and Machurian, are all a part of an Altaic superfamily — along with Japanese, Turkish, and a number of others. This was broadly accepted fifty years ago, and less so today.