Fletcher, 21, has left the Hammers' training camp in Germany for talks with the Teessiders.
The forward has scored one goal in 20 games since moving from Manchester United in the summer of 2016.
He scored eight goals in 27 games helping Barnsley win promotion to the Championship in 2015-16.
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Middlesbrough have agreed "a deal in principle" to sign West Ham United striker Ashley Fletcher, although terms are currently unconfirmed.