The former Arsenal and Manchester City ace's 18-month suspension was up last week.
And The Mirror reports Manuel Pellegrini is eyeing a swoop, with Everton chief Marco Silva also interested.
Nasri, 31, was banned after pictures emerged of him taking an intravenous drip in 2016.
But he is now free to play and could be registered for a Premier League club in January.
The Frenchman would bring plenty of experience and represents a bargain for a player of his calibre.
Nasri won the Premier League with Pellegrini at the Etihad and the duo could link up again.
Manuel Lanzini, Andriy Yarmolenko, Carlos Sanches and Jack Wilshere are all struggling with injury and the Hammers need reinforcements.
While Nasri must wait to play in matches, he is now free to train for clubs and get fit in time for the New Year.
Everton could yet launch a bid as they look for a creative spark to work alongside Richarlison and Gylfi Sigurdsson
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