In August, Uefa won an appeal for the ban to be increased to 18 months, backdated from July 1, 2017, although Nasri has played since then.
The story suggests that Nasri, who played under Pellegrini at Manchester City, is undergoing a medical prior to signing a short-term contract with the Hammers.
However, according to The Mirror, West Ham have not been left encouraged by the playmaker's current fitness levels and there are doubts over whether Nasri will be available to play in January.
Nasri had the drip treatment at a Los Angeles clinic two years ago while he was on loan at Sevilla from City during the 2016/17 season, prompting the Spanish anti-doping agency to open an investigation.
Donald Trump ignores Emmanuel Macron 'nationalism' dig, honors veterans in France
Sunday, Macron rebuked nationalism in perhaps another dig at Trump who has been vocal about his " America First " agenda. At the forum German Chancellor Angela Merkel warned against taking peace for granted, saying, "We have to work for it".
The Hammers are now missing Jack Wilshere, who signed from Arsenal in the summer but fell victim to yet another injury, and Andriy Yarmolenko, who tore his achilles tendon three weeks ago, ruling himself out for the remainder of the season.
Nasri will be free to resume his playing career once the ban ends at the end of 2018.
Everton manager Marco Silva had discussed a move for the player with his backroom staff but Nasri did not fit the age profile of players that the Merseyside club want to buy.
It is understood that West Ham will bow to wage demands of £80,000 a week, or £5m a year, by Nasri.