The 23-year-old joined the Welsh outfit from Ipswich Town for around £1.5m last summer and went on to make 39 appearances across all competitions.
One previous report has claimed that Crystal Palace were closing in on the signing of Downes, while Southampton and Wolverhampton Wanderers have also been linked with the midfielder.
However, Sky Sports News journalist Dharmesh Sheth reports that West Ham are now "increasingly confident" that they can complete the signing of Downes for a fee believed to be worth £12m.
The England Under-20 international has spent his entire career in the EFL but he now looks set to make the step up to the Premier League and could become West Ham's third signing of the summer after Nayef Aguerd and Alphonse Areola.
It Is thought that David Moyes is also keen to re-sign Jesse Lingard, who is now a free agent, while a striker is also high on the Scotsman's agenda amid links with the likes of Chelsea's Armando Broja and Sevilla's Youssef En-Nesyri.