The deal is for $870,000 with no signing bonus, according to ESPN's Ed Werder. Johnson could earn up to $2 million by rushing for 1,300 yards and being named to the Pro Bowl.
Fox Sports first reported the agreement.
The Cardinals reached a deal with Johnson because of injuries to starter Andre Ellington (hamstring) and the next two players on the depth chart. Ellington has a troublesome injury history that suggests the Cardinals not rely totally upon him.
Johnson was chosen over other veteran options because the Cardinals view him as having a skill set similar to Ellington's -- both are good receivers -- making them interchangeable in coach Bruce Arians' offense.
Johnson, who will turn 30 in September, became an unrestricted free agent after the New York Jets declined to exercise their option following the 2014 season. Johnson recorded career lows in carries (155), rushing yards (663) and rushing touchdowns (one) last season, after signing a two-year, $8 million deal. It was the first time in his seven-season career that Johnson failed to rush for 1,000 yards in a season.
Before his 1,077-yard season with the Titans in 2013, in which he averaged 3.9 yards per carry, Johnson had never averaged fewer than 4.0 yards per rush in a season. Even in 2014, he gained 4.3 yards per rush.
By comparison, the Cardinals averaged 3.29 yards per rush last season -- the lowest in the NFL.
In January, Johnson was arrested in Orlando, Florida, for openly carrying a weapon, but officials decided not to pursue charges. In March, he was shot in the right shoulder during a drive-by shooting that left a friend with bullet wounds and the driver of their car dead.
The Cardinals were thin at running back the first two weeks of training camp. Both starter Ellington and third-round pick David Johnson were out with hamstring injuries. Ellington returned late last week and played in Arizona's preseason game Saturday, while Johnson returned to practice Monday but was limited.
On Monday, before the Chris Johnson deal was reported, Arians said reports of Arizona pursuing Johnson might not have motivated Ellington to return, but it was too early to tell if such reports helped hasten David Johnson's return.
"We'll see with David," Arians said. "He's so young that he probably doesn't even know who Chris Johnson is."