The 19-year-old has been compared to Bayern Munich midfielder Thiago Alcantara mainly because of their identical style of play and his huge talents.
City boss Pep Guardiola and United boss Ole Gunnar Solskjaer are set to do battle over him, but the Hammers could win the race if the teenager is convinced of a quicker path to first-team football.
Manager Manuel Pellegrini wants to turn West Ham to a top-seven side and is looking to improve the quality of his squad.
Elmkies will definitely be linking up with the under-23’s if he is snapped up, but he has the potential to quickly force his way into the Chilean first-team plans sooner rather than later.
The Israel under-19 international could cost just £2.5 million this summer, but will be available for just training compensation next summer if he doesn’t sign a new contract with only a year left on his deal.
Elmkies impressed for Hoffenheim in the UEFA Youth League last season, particularly against City, and there are plenty suitors keen on his signature too.
West Ham have shown they are open to giving youth a chance, with Grady Diangana, 21 and Declan Rice, 20, featuring heavily last term, and the Israeli youngster could fancy his chances at the London Stadium ahead of a move to either United or City.