Brighton, who finished second in the Championship last term, have been working hard in the summer transfer window ahead of their debut Premier League season and four new faces have arrived at the AMEX Stadium.
Pascal Gross, Markus Suttner, Isaiah Brown and Matthew Ryan, who cost the Seagulls a club-record £5.1million, have been added to the ranks and more signings are expected before the window shuts.
Manager Chris Hughton is said to be showing a keen interest in Lawrence, who has failed to cement down a regular first team place since arriving at the King Power Stadium in 2014.
The Wales international spent last season on-loan at Ipswich Town and, despite featuring in pre-season ahead of the new campaign, he is expected to head for the exit door this summer.
Brighton are ready to splash the cash on the 23-year-old with the club preparing an £8m offer - an amount that would smash their previous highest fee spent on Ryan earlier in the window.