The Clarets have had a quiet transfer window and have made just one outfield signing in centre-back Nathan Collins from Stoke City.
They are likely to strengthen their attack this month and have their sights on pursuing Cornet, who has been with Lyon since 2015.
Their opening bid of £12 million plus add-ons has bettered Hertha Berlin’s offer, but they are still short of meeting the £15m price tag.
The Clarets are still considered favourites to sign him with no improved offers thus far.
Cornet is a winger by trade, and he prefers to play on the right side of the attack.
However, he has been converted into a left-back over the past 12 months, where he has impressed with his tackling and ability to deliver key passes.
In the 2020/21 campaign, the 24-year-old bagged four goals and six assists from 39 outings. The majority of those contributions came from left-back.
Hertha are reportedly the Clarets’ main competition, but they may need to act fast to secure an agreement with Leeds United and Watford keeping tabs.
The Lancashire outfit could sign more than one attacker this summer. They have their sights on re-signing Aaron Lennon, who is a free agent.
Stats from Transfermarkt.com