The club are close to agreeing personal terms with the Ivory Coast winger and it is hoped he will undergo a medical on Friday morning, with his signing potentially confirmed ahead of the weekend.
Cornet joined Burnley from French side Lyon last summer in a deal worth £12.5million, and was one of their best performers last season.
However, his nine Premier League goals were not enough to keep the Clarets in the top flight and the 25-year-old was still expected to leave following the club’s relegation. He did not feature in Burnley’s Championship opener against Huddersfield last Friday, with manager Vincent Kompany claiming he was not physically ready.
Cornet has been used primarily as a winger by Burnley but is seen by David Moyes as an option at left-back, as well as further forward.
He would become West Ham’s fifth signing this summer and provide a timely boost to the Hammers in the face of frustration over moves by Filip Kostic from Eintracht Frankfurt and Lille midfielder Amadou Onana which failed to materialise.
The Irons are struggling to convince Kostic to quit Champions League football with the German side and are unwilling to meet Onana’s wage demands, despite having agreed a fee with Lille.