The South London Press are reporting that the Clarets have had a bid accepted for Charlton Athletic's Iceland international Johann Gudmundsson.
The offer is thought to be in the region of £2.5m with another £1m in add-ons.
The 25-year-old midfielder missed just four minutes of his nation's incredible Euro 2016 campaign in France as Heimir Hallgrímsson's underdogs reached the quarter-finals where they were beaten by the hosts.
The Valley favourite, who claimed an assist in Iceland's 1-1 draw with Portugal in Group F at the Stade Geoffroy-Guichard, was on the scoresheet six times for the Addicks in the Championship last term and added 11 assists.
Gudmundsson, who impressed on the final day of the 2015/16 campaign as the Clarets secured the title, was expected to move on following the club's relegation to League One.