The Hammers are keen to offload Yarmolenko during the ongoing transfer window as they look to reduce his £115,000-a-week salary from their wage bill.
It was recently reported that Fenerbahce could sign the Ukrainian for a cut-price £2 million fee, but it appears that they could face competition from the Clarets.
Burnley are aiming to reinforce their attack and are prepared to bid around £3m to bring the 31-year-old to Turf Moor this summer.
Yarmolenko has struggled with injuries during his three seasons with the Hammers, and he has not been able to secure a regular starting role.
Overall, he has registered a tally of 10 goals and six assists from 54 appearances. He was a regular unused substitute earlier this year.
Despite this, he made a huge impression for Ukraine at the European Championships, bagging two goals and two assists from five outings.
Yarmolenko may want another chance to prove himself in the top-flight, and the Clarets could offer him the ideal platform with consistent playing time.
He may have to take a significant pay cut for the deal to go through. James Tarkowski and Chris Wood are the highest earners at Burnley at £50,000-a-week.
Stats from Transfermarkt.com