A free agent, Nasri undertook a medical at the request of the club, after they were unsure of offering a contract to the out-of-practice midfielder.
The Frenchman is nearing the end of an 18 month ban for doping, after he was sanctioned by UEFA for undergoing an intravenous drip treatment while on holiday in the United States two years ago.
West Ham manager Manuel Pellegrini is keen to be reunited with Nasri, with whom he tasted Carabao Cup and Premier League success while both men were at Manchester City.
However, after the player demanded a £95,000 per week contract, the 'Irons' have since counter offered, by giving the player until the new year to prove his fitness and thus earn his place with the Premier League side.
The East London club have also asked that Nasri lower his wage demands, in light of the fact that he has not played for almost two years. Nasri is allowed to train, but cannot be selected until his ban ends in January.
Midfield options have been thin for the 'Irons', who have a long injury list at the middle and top ends of the park. Yet Nasri is said to be at least two months off full match fitness, so West Ham will have to hope that he comes good if they do decide to offer him a deal.