The Hammers blamed the late announcement of the double swoop - nearly a day after the transfer window closed - on administrative issues but the pair are expected to be passed eligible to play in England.
Vastly-experienced Di Michele, 32, has been capped six times by Italy and has also starred for Palermo and Udinese while 26-year-old Ilunga made 136 appearances for St Etienne before joining Toulouse in 2007, where he was a regular last term.
Ghana midfielder Appiah, meanwhile, is in talks with the club about a securing a contract and can join the Hammers outside of the transfer window as the 27-year-old is a free agent after being released by Turkish side Fenerbahce recently.
A club statement read: "West Ham can confirm discussions with Stephen Appiah are ongoing."



