The consortium will pay 20 million pounds for a 98.5 percent stake in the club from current owner Patrick Cryne, who is terminally ill and said to fans he was "living on borrowed time" in a poignant letter earlier this week.
Chinese businessman Chien Lee, who is majority shareholder at French top tier club Nice, would become the majority shareholder of the English club if the deals goes through. American businessman Conway is also a director at Nice.
Billy Beane, a former American baseball player portrayed by U.S. actor Brad Pitt in the hit movie "Moneyball", will take a 10 percent stake in the club, the source said. Beane is a minority owner of U.S. baseball team Oakland Athletics.
Reuters could not immediately reach Lee, Beane or Barnsley for comment.
Chinese investment in European soccer - which boomed last year amid state support - has lost some of its glitz after a crackdown by China's government on capital outflow amid concern over depreciation of China's currency and excess capital flight.
Authorities have been cracking down so-called "irrational investment", particularly in sectors such as real estate, entertainment and sports.
The club, situated in a former coal-mining town in South Yorkshire, is currently in the second tier of English football, where it has spent most of its 130 years of existence.
(Reporting by Pei Li in BEIJING and Adam Jourdan in SHANGHAI, editing by Nick Mulvenney)
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