The Hammers boss was seen in takes with Juve director Fabio Paratici and reports in Italy suggest he hopes to sign the pair.
Dutch winger Elia has been described as a 'flop' after one season in Turin and he is desperate to leave the Serie A champions.
Krasic is seen as the least likely of the two players to move to Upton Park, having impressed at the start of his Juve career, he fell out with boss Antonio Conte last season.
West Ham are also close to sealing a deal for Bolton goalkeeper Jussi Jaaskelainen.
The Finnish keeper ended a 15-year stay at Bolton and is set to replace Robert Green, who looks likely to move across London to Queens Park Rangers.