Lawrence, a forward who scored 11 goals in the Championship last season, is expected to leave the Rams this summer following their relegation to League One.
The Wales international is also a target for West Brom, Watford and Rangers, according to reports.
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Having started his career at Manchester United, Lawrence joined Leicester City in the summer of 2014 following their promotion to the Premier League before gaining Championship experience while on loan at Rotherham United, Cardiff City, Blackburn Rovers and Ipswich Town.
A fine 2016/17 season with the Tractor Boys saw Lawrence win the club’s Player of the Year and Players’ Player of the Year awards for his performances, as well as the Goal of the Season gong for a strike against QPR.
That also earned him a move to Derby in the summer of 2017 for a reported fee of up to £7 million.
He has since made 185 appearances for the club, scoring 37 goals.
Lawrence won the last of his 23 Wales caps in March 2021 and his experience in the second tier stretches more than 260 games.
The 28-year-old was made Derby’s captain by former boss Wayne Rooney last summer and subsequently enjoyed his best-ever goals return in a Rams shirt in the 2021/22 campaign.
Sheffield United have also been linked with Hibernian left-back Josh Doig, former Crystal Palace midfielder Nya Kirby and centre-half Anel Ahmedhodžić of Swedish side Malmö in recent days.
Blades boss Paul Heckingbottom is yet to make a signing this summer, however.