Sources told ESPN that the deal is worth up to $4 million.
Smith, 30, has not played in a game since he was suspended in 2015 because of legal and substance abuse issues, but he is in the process of seeking reinstatement to the NFL, according to multiple sources.
He posted a pair of photos to Instagram on Wednesday night, including one of him signing the contract. In the caption he wrote, "Life is good. I'm thankful. I'm blessed. I'm a Cowboy."
The second photo Smith posted was a statement, in which he wrote in part, "I've learned how to take a different perspective on the adversities of life. Instead of looking at life as a victim, I have embraced the journey as God has planned it, making exponential strides toward becoming a better man."
The contract does not include any guaranteed money, but Smith will earn $90,000 once he is reinstated by the NFL. Thirty days after reinstatement, he will get another $50,000. When he shows up for training camp, he will receive $100,000, another $100,000 after two preseason games and $100,000 at the end of the preseason. If he does not make the 53-man roster, the Cowboys would be on the hook for $440,000.
Smith will earn a $910,000 base salary and $40,625 every time he is on the active gameday roster, totaling $650,000. He has incentives totaling $2 million for sacks -- $500,000 for eight, $1 million for 10, $1.5 million for 12, and $2 million for 14.
The seventh overall pick in the 2011 draft by the San Francisco 49ers, Smith has recorded 47.5 sacks in 59 career games.
Cowboys owner and general manager Jerry Jones has a history of taking chances on players with difficult pasts.
In 2014, he traded with Baltimore for linebacker Rolando McClain, who led the team in tackles that season. In 2015, he signed Greg Hardy as a free agent and selected Randy Gregory in the second round of the draft. A decade earlier, he added Tank Johnson and Pacman Jones.
In 1992, the Cowboys traded for defensive end Charles Haley, who had issues in San Francisco and helped the organization win back-to-back Super Bowls.
Part of the allure of Smith is his connection to the Cowboys' new coaching staff. Smith had his best two seasons under defensive line coach Jim Tomsula with the Niners, including a 19.5-sack season in 2012, when he was named an All-Pro.
Tomsula was one of the first additions to Mike McCarthy's staff after Mike Nolan was named Dallas' defensive coordinator.
Having high expectations for a player who last played in a game on Nov. 15, 2015, might seem foolish, but the Cowboys have a need for pass-rush help. Robert Quinn, who led the Cowboys with 11.5 sacks in 2019, left as a free agent for the Chicago Bears.
Gregory, who was suspended indefinitely in February 2019, has started the reinstatement process and would be welcomed back, even though he has played in just 16 games since his rookie season. He finished second on the Cowboys in sacks in 2018 with 6.5, after missing 30 of 32 games.
Smith had his first off-the-field incident in 2013, when he was arrested on suspicion of DUI after crashing his truck into a tree in a residential neighborhood in San Jose, California. In 2014, the NFL suspended him nine games for violating the league's personal conduct and substance abuse policies.
In August 2015, Smith was released by the Niners the day after he was arrested on hit-and-run, DUI and vandalism charges. He signed with the Oakland Raiders a month later but was suspended for a year for violating the league's substance abuse policy.
The Raiders signed Smith to a two-year deal in 2016, but released him in 2018 after he was arrested on a number of charges including domestic violence, assault and false imprisonment. Smith pleaded no contest to a pair of misdemeanor charges in November 2018 to settle his domestic violence case from earlier that year.
The Cowboys are hoping that a reunion with Tomsula and a renewed commitment to sobriety can get Smith's career back on track.
ESPN's Paul Gutierrez contributed to this report.