It is anticipated that the 30-year-old former York City, Manchester United and Middlesbrough star will seal a permanent move to the Cottagers next summer.
Greening has spent the last five seasons at The Hawthorns, featuring in over 200 first-team matches for the Baggies.
Roberto Di Matteo: "Jonno has been a big player for this club and, as you well know, we made every effort to keep him.
"But eventually we felt the deal on the table was a very good deal for us and we decided to let him go. We wish him good luck as he's been a great servant for this club.
"Since I've worked here, he's been very professional and I wish him all the best in the Premier League."
2009/10 BORO RESULTS
MACCLESFIELD 0 BORO 3 (Arca pen, Emnes, Franks)
(1st half) Coyne, Hoyte, Huth, Williams, Grounds, Yeates, Walker, Digard, Cronesberry, Aliadiere, Martin. (2nd half) Coyne, McMahon, Bates, Hines, Johnson, Smallwood, O'Neil, Arca, Porritt, Emnes, Franks.
DUMBARTON 0 BORO 5 (O'Neil, Dunlop og, Aliadiere pen, Johnson, Williams)
Jones (Coyne 46), Hoyte (McMahon 72), Bates (Wheater 72), Huth (Hines 72), Grounds (Taylor 72), O'Neil (J Johnson 72), Arca (Capt) (Williams 72), Digard (Walker 29), Yeates (A Johnson 65), Aliadiere (Franks 46), Emnes (Porritt 63).
CARLISLE 0 BORO 2 (Johnson, Williams)
Jones (Coyne 60),McMahon,Taylor (Williams 72),Bates(Walker 24),Huth,Wheater (Hines 60),O'Neil,Arca(Hoyte 60) 60),Franks,Yeates (Porritt 72),A Johnson (Martin 84)
OLDHAM 2 BORO 2 (Williams, Huth)
Coyne; Williams, Wheater (Huth 64), Hines, A Taylor; Hoyte (McMahon 64), Walker (Yeates 54), Arca (Smallwood 79), A Johnson (Martin78); Aliadiere(J Johnson 78) Franks (Porritt 78) Substitutes (not used): Ripley
MILLWALL 1 BORO 1 (Williams)
Coyne (Jones 46), Hoyte (McMahon, 61), Huth, Wheater, Ground, Yeates, Arca (O'Neill 46), Digard (Williams, 61), Johnson, Aliadiere, Emnes (Franks, 81) Subs not used: Adan Smith, Seb Hines , Andrew Taylor, Josh Walker, Nathan Porritt Goals: Wheater OG 27mins, Williams 78m.
BORO 0 SHEFF UTD 0
Coyne, McMahon, Wheater, Huth, Grounds, Yeates,Williams, Digard (O'Neil 42), Adam Johnson, Aliadiere,Emnes (Lita 65).
Subs Not Used: Steele, Hoyte, Taylor, Arca, Franks.
SWANSEA 0 BORO 3 (Johnson, Emnes, Tuncay)
Coyne, McMahon, Wheater, Huth, Williams, Grounds, Yeates (Arca 85), O'Neil, Adam Johnson, Lita (Franks 79), Emnes (Sanli 67).
Subs Not Used: Steele, Hoyte, Taylor, Hines.
SCUNTHORPE 0 BORO 2 (Johnson 2, 1 pen)
Coyne, McMahon, Williams, Huth, Grounds, Yeates (Arca 84), O'Neil, Wheater, Adam Johnson (Hoyte 90), Lita (Sanli 74), Emnes.
Subs Not Used: Taylor, Hines, Franks, Steele.
BORO 2 DONCASTER 0 (Tuncay, Lita)
Coyne, McMahon, Wheater, Huth, Grounds,Yeates (Aliadiere 72), Williams, Arca, Adam Johnson,Lita (Taylor 84), Emnes (Sanli 16).
Subs Not Used: Steele, Hoyte, Hines, Bennett.