Fletcher, 20, spent the second half of last season at Oakwell, scoring in the play-off final as Barnsley won promotion to the Championship.
Leeds have been monitoring Fletcher throughout the summer and were encouraged by his decision to turn down a new deal at Old Trafford last week but West Ham have killed competition for his signature by offering the 20-year-old a four-year contract.
Barnsley may still be able to lure Fletcher away on a loan deal this coming season, but it is understood Leeds were more interested in a permanent move.
Fletcher is out of contract at Manchester United but the club are entitled to compensation for him on account of his age. He is expected to cost in the region of £500,000.
Fletcher initially planned to start pre season at Old Trafford while he weighed up his options but West Ham have forced the issue by stepping in with a substantial proposal.
He is expected to undergo a medical with West Ham later today.