Barr will sign a five-year contract worth $67.5 million with $33 million in guarantees.
Barr has spent all five of his National Football League seasons with Minnesota, earning Pro Bowl nods in each of the past four years.
NFL Media first reported Barr's change of heart. Jordan verbally agreed to terms on a deal with Dallas before returning to Los Angeles.
The 26-year-old pass rusher has made the Pro Bowl in each of the last four seasons with the Vikings.
Defensive end Everson Griffen, whose $10.9 million base salary for 2019 becomes fully guaranteed Friday, remained the most obvious candidate for a restructured contrac; sources said the Vikings have submitted an offer to the defensive end that would rework the final four years of his deal and help the team clear cap space.
A nagging hamstring injury limited Barr to 13 games in 2018, his fewest since his rookie season. Barr was New York's top option at the position, but there are still plenty of talented edge rushers still available for the taking.
"It's been a insane couple days, but I'm where I want to be", Barr wrote on his Instagram page.
There was a hint that Richardson would not re-sign when the Vikings agreed Monday to a three-year, $12.45 million contract with Shamar Stephen, who played for them from 2014-17 and spent last season with Seattle.