Two days after former Pittsburgh Steelers running back Willie Parker visited with Washington, the Redskins are scheduled to host a visit Thursday with Johnson, formerly of the Chiefs and Bengals, the source said.
The Redskins have made it known around the league that they would like to sign a veteran back.
Johnson, whose production has declined in recent years, appeared in seven games for both the Chiefs and Bengals last season, totaling 178 carries for 581 yards.
His career in Kansas City ended midway through its seventh season when the team let Johnson go as he was set to return from his second suspension in a 12-month period because of bad behavior.
Information from The Associated Press was used in this report.