The on-loan Aston Villa midfielder suffered the injury against Burnley last Saturday, and will be out for at least the next fortnight.
That would rule the 21-year-old out of the clash with Wolves this weekend, however, with an international break the following week Davies is hopeful Osbourne will be fit for the game against Barnsley on April 4.
"He was scanned on Tuesday and it's not as bad as we first thought, it certainly won't be months," the manager told the club's official website.
"We're probably looking at two or three weeks but the good news is the
fact that it's not as damaged as we thought it might have been.
"We're coming up to an international break which will give him time to get back but on the whole it is positive news."