The 22-year-old has been restricted to five appearances for the Iron this season, but would have started against Oxford in place of the suspended Sam Winnall had it not been for this latest setback.
"Matty is going to be out a few weeks with that," boss Russ Wilcox told the club's official website.
"It's not a good injury and it's very swollen around his ankle. We think it's ligaments. He's had an x-ray and there's no break in there, which is great news."
Godden recently returned from a loan spell with Dartford, where he scored five goals in nine appearances.
Wilcox added: "It's probably going to be four to six weeks. It's a bit blow for Matty.
"It's not quite happened so far for Matty Godden at this football club for whatever reason.
"He would have had his chance (on Sunday). I would have definitely started him up front with Deon.
"He's lost that opportunity, but hopefully we can get him back as soon as we possibly can."