The Spanish midfielder is seen as having great potential for the future but he didn't set Serie A alight last season with Sampdoria.
Obiang is reportedly keen to find a fresh challenge but Sampdoria have been very reluctant to sell him and their valuation is said to have put off several Italian clubs, including AC Milan.
West Ham's fellow Premier League sides Queens Park Rangers and Crystal Palace are believed to be interested in the Atletico Madrid youth product and that may have forced the East London club into putting down an offer.
West Ham would pay just £800,000 initially, effectively acting as a paid season loan, and then follow that up with £4.8 million in June 2015.
However, Sampdoria sporting director Carlo Osti reportedly remains unwilling to let Obiang leave, so £800,000 this summer may be somewhat underwhelming.
Do West Ham need Pedro Obiang?