He is not due to be back in action until next season and the East Yorkshire club last week confirmed he would be moving on elsewhere.
Salford today announced they had signed Greenwood on a two-year deal and, having reached an agreement with KR, the recuperating forward would switch over immediately.
The Red Devils signed England Knights half-back Atkin last week for 2020 and Greenwood said: “Salford are doing really well at the moment.
“This is a new step for me and I’m really looking forward to it.
“Being around the boys, the conditioners and the coaches now, means I’ll be able to make that transition straight away.
“I’m just concentrating on getting myself back on that field, then everything else I’ll take care of once I get to that point.”
Oldham-born Greenwood joins his Hull KR teammate Chris Atkin at the AJ Bell Stadium, who has also agreed a two-year-deal with the club.
Greenwood played more than 80 games for KR after initially joining in 2015 and added: “It’ll be good to link up with Chris again.
“We had a combination going over there in certain games.
“I know how he plays and he knows how I play, so it should be good for the team.
“I’m really excited for next year.”
Greenwood has experienced playing with Salford before, having joined the club on loan from Wigan Warriors during the 2015 season, scoring a try against Hull FC.
Head coach Ian Watson said: “Greeny’s a great bloke and a big, no-nonsense, mobile forward, who’s is equally effective in the middle, or second-row.
“He’s got good leg speed and footwork. He runs strong lines, with great ball playing abilities and an attitude in defence to match.
“At 28, he’s coming into his best years and I believe we’ll see them at Salford.”