NFL Network Insider Ian Rapoport reported that both players likely will not receive long-term deals with their respective teams prior to Monday's deadline, "barring a significant turn of events."
If deals are not signed and approved by the league before 4 p.m. ET, the sides cannot agree to new contracts until after the regular season is over.
Lawrence, who has been seeking a long-term contract from the Dallas Cowboys, is set to play for just over $17 million in 2018 with the franchise tag.
The 26-year-old Lawrence started all 16 games last season and amassed a career-high 14.5 sacks.
Ansah recorded a team-high 12 sacks, 44 total tackles, one forced fumble and one fumble recovery last season with the Detroit Lions despite missing two games with a back injury.
The 29-year-old Ansah has registered 44 sacks over his first five seasons, including 14.5 sacks in 2015. He had just two sacks the following year while being bothered by a high ankle sprain that limited him to 13 games.