Very likely a loanword from Altaic. S. A. Starostin's Altajskaja problema i proisxozhdenie japonskogo jazyka [The Altaic problem and the origin of the Japanese language] (Moscow, 1991), p. 69, reconstructs as follows:

Altaic *mü:rki 'barley [jachmen'], wheat [pshenica]'

• Tungus-Manchu *murgi 'barley [jachmen']'
• Middle Korean mír(h)- 'wheat [pshenica], barley [jachmen']'
• Proto-Japanese *mùnkí 'grain [xleba], cereals [zernovye]'
  • Old Japanese mugi
  • Ruiju meigishoo mùgí
    [The Ruiju meigishoo is a Japanese dictionary of about 1100 C.E. which records the accentual system of a dialect of that time.]

In a footnote (no. 136, p. 138), after specifying the various Tungus-Manchu forms, Starostin includes a comment on Turkic and Mongolian forms:

"The question of the relationship to these forms of Turkic *bug-taj 'wheat [pshenica]' (< *burg-taj < *mürk- ? -- cf., with -r-, Chuvash po[breve]ri, Tatar, Kazakh bu[breve]raj 'spelt [polba]' -- is a complex one. Written Mongolian buQudai ["Q" = lower-case gamma in "ipa-ascii"--WB] 'wheat [pshenica]' is most likely from Turkic; cf. M. Räsänen, Versuch eines etymologischen Wörterbuchs der Türksprachen (Helsinki, 1969), 86, and È. V. Sevortjan, Ètimologicheskij slovar' tjurkskix jazykov (1974-1980), 233-235."

In the same note, Starostin adds:

"It is interesting to note the likely connection of this root with Old Chinese *[mr@:k] 'barley [jachmen'], wheat [pshenica]'."

The new version of my Old Chinese reconstruction (which follows Starostin's ideas on vowel length), is identical to Starostin's for this word. Note that the vowel length in Old Chinese *[mr@:k] matches the vowel length reconstructed by Starostin for Altaic. Also, although the *r follows the vowel in Altaic, it precedes the vowel in Old Chinese. This is an adaptation to Old Chinese syllable structure: Old Chinese (as presently reconstructed) does not allow clusters like *rk after the main vowel.