Lawrence has been seeking a long-term contract but is set to play for just over $17 million in 2018 after the Cowboys slapped the franchise tag on him in March.
The clock is ticking on a deadline to reach a new term. All franchise-tagged players have until July 16 to sign a long-term contract with their current team.
Lawrence, who started all 16 games and amassed a career-high 14.5 sacks in 2017, is seeking a five-year deal, the Athletic reported, citing a source.
A second-round pick of the Cowboys in the 2014 NFL Draft out of Boise State, Lawrence earned his first Pro Bowl berth last season after tying Calais Campbell of the Jacksonville Jaguars for the second-highest sack total in the league.
The 26-year-old Lawrence's breakout season occurred after he underwent back surgery for the second straight offseason in January 2017. In four seasons with the Cowboys, Lawrence has 23.5 sacks and 133 tackles.