Jordan is currently on the books of Aston Villa, but is said to be keen on a move to the Hammers following his club's relegation to the English Championship.
The 24-year-old, who is valued at £8 million by the Villans, played alongside his brother during their time together at Marseille, and is open to the idea of doing so once again at West Ham.
Andre, meanwhile, is the said to be the subject of a £16 million Hammers offer, with his club, Swansea, already lining up a replacement in the shape of Sevilla forward Fernando Llorente.
The offer for Llorente has been confirmed by Sevilla's director of football, Monchi, who told the club's official website: "It has come a long way with Swansea to reach an agreement, we expect to hear in the coming days."
While a deal for the Ayew brothers is seemingly close to be being completed, Bilic also has other transfer targets ahead of the new Premier League campaign.
Left-back Davide Santon, who represented Newcastle United between 2011 and 2015, is wanted by the Croatian coach following the long-term injury suffered by first-choice Aaron Cresswell.