Greenwood, who did well in pre-season and played against Everton for the full 90 minutes at Hillsborough is set to make his debut for The Owls in their league cup tie.
The in-form Richard Wood has been suffering with a toe problem, Guy Branston has pulled a calf muscle and reserve Chris Carr has tendonitis so Smith's options are limited.
Greenwood will fit in at right-back while Patrick Collins and Lee Bullen will play in the middle with perhaps Paul Smith coming in at left-back because of the injury to Paul Heckingbottom.
The Owls are still missing Matt Hamshaw and Graeme Lee although they will be back next month.
Meanwhile, Rotherham boss Ronnie Moore has hinted that Paul Sturrock will be the new manager at Hillsborough next season by saying: "If I was a betting man, which I am, I would put my house on Paul being the new manager at Wednesday."
Moore has been one of the front runners for the role in the past and has been linked this time around but fans will have to wait until tomorrow to find out whether Sturrock is confirmed or not, with every media source seemingly pointing the finger his way.
There is still time for Gary Megson to resign from West Brom though: his Premiership team lost 2-1 to Colchester last night!