It is understood the Hammers will raise their offer for the player who began his career at the east London club.
Defoe, 34, has scored 11 goals in 21 appearances for the Premier League strugglers this season.
His latest goals came on Monday as Sunderland twice fought from a goal down to draw 2-2 with in-form Liverpool, which left manager David Moyes’s side in 18th position.
Defoe’s senior career began at West Ham before he moved to rivals Tottenham in a £7m deal in 2004, with striker Bobby Zamora going the other way. A £7.5m move to Portsmouth followed in January 2008 before he returned to Spurs the following year for £15m.
Defoe made a surprise move to Canadian team Toronto FC in 2014 before he was lured to Sunderland in 2015 by former Black Cats boss and ex-Spurs team-mate Gus Poyet.
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