The 24-year-old dual-premiership Tiger toured the Power's Alberton Oval headquarters in recent days and had been weighing up a lucrative offer to return to his home state.
But he has now reportedly told Richmond officials he will not request a move when the annual trade period opens on Monday.
Graham is contracted for 2023 and has a trigger that will extend the deal by a further season.
He will be a free agent at the end of 2024.
Graham's decision is a boost for the Tigers as they look to regenerate their list for another premiership push after falling short the last two seasons.
Fellow midfielders Tim Taranto and Jacob Hopper are expected to leave GWS for Richmond, while Giants teammate Tanner Bruhn has been linked with Geelong.
The Cats are also hoping to strike a deal with Collingwood for talented young forward Ollie Henry.
Meanwhile, Port Adelaide remain keen to boost their forward line firepower with the acquisition of West Coast goal sneak Junior Rioli.
The Power have also met with versatile Geelong tall Esava Ratugolea, who failed to hold down a regular spot as the Cats marched to this year's premiership.
The 24-year-old ruck-forward managed just four senior games, taking his career tally to 59, and was trialled in defence at VFL level.
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