Konate, a striker who can also play on either wing, scored 14 goals for French side Amiens as their first ever Ligue 1 season ended with a mid-table finish in 2017/18.
He came off the bench for Senegal in their 2-1 win over Poland in Moscow on Tuesday, and he could be subject to a bidding war when he returns from the World Cup, with the Sun reporting that Sean Dyche and Manuel Pellegrini are both interested in adding him to their squads.
West Ham could have the edge, as Senegal captain Cheikhou Kouyate can sound out Konate to see if he would be interested in joining him at the London Stadium.
On the other hand, Burnley can offer Konate European football as they will play in next season's Europa League. Wednesday's draw saw them paired with Aberdeen in the second qualifying round.
Konate scored against Great Britain in the 2012 Olympics football tournament, and he has plied his trade in a variety of countries including Israel, Russia, Italy and Switzerland since leaving Senegal in 2011.
It has already been a busy week for West Ham in the transfer market. Defender Issa Diop completed a £22m move from Toulouse on Tuesday, and the club confirmed the signing of goalkeeper Lukasz Fabianski from relegated Swansea on Wednesday.
They will officially join the club on the first day of July, along with Ryan Fredericks, who signed on a free from Fulham.