It has been rumoured that the 21-year-old would be moving to Hull City for a fee of around £1.5m, but it seems the Rams may make a late move for the player.
According to The Sun, Burke was set to move to Yorkshire club Hull City after flying back from West Ham's training camp to undergo a medical, but Derby's improved offer for the player may have complicated the situation.
The player is said to be more interested in Derby due to manager Lampard's connection to West Ham, where Burke has been since he was a youth player.
He made 25 league appearances last season as the Trotters avoided relegation on the final day of the season, before returning to West Ham.
Derby will be hoping to improve their squad as they bid to better their sixth placed finish last season, when they lost to Fulham in the semi finals of the play-offs.
Lampard has since taken over from Gary Rowett, and has already snapped up former Arsenal forward Aaron Eyoma on a free transfer.
Hull meanwhile will remain hopeful they can conclude negotiations with Burke, as they look to forget their first season back in the Championship following relegation. The Tigers finished 18th, just eight points clear of the relegation zone.