The Welshman captained Wayne Rooney's Derby during a turbulent campaign for the East Midlands club, which has seen them relegated from the Championship following a total 21-point deduction as a result of their financial difficulties.
Despite this, the 28-year-old still managed to net 11 goals this season, and this form, as well as Derby's relegation, has resulted in a number of clubs stepping up their interest according to the Daily Mail.
The report suggests that Burnley are one of the sides said to be interested in acquiring Lawrence's signature this summer, as well as Norwich City, Watford and Fulham.
Whilst the Clarets remain uncertain which division they will be playing in next season, Norwich and Watford will drop into the second tier following their confirmed relegations from the Premier League, whilst Fulham will move in the opposite direction after earning promotion from the Championship.
With the former Manchester United academy product seeing his contract expire at Pride Park in June, a free transfer move for the forward has attracted plenty of interest.
Since arriving at Derby from fellow East Midlands side Leicester City, Lawrence has gone on to make 185 appearances for the club during his five-year stay.