The 30-year-old, whose 18 goals helped fire the Millers to League Two promotion last season, suffered a serious-looking hamstring injury in Saturday's friendly defeat toHuddersfield.
But scans show the problem is not as bad as first feared, with the former Manchester United man expected to return to training next Monday with an eye on the Millers' opening game at Gresty Road.
Assistant boss Paul Raynor said: "Hopefully Daniel will start training next week and be available for Crewe, so that's very positive that it wasn't as serious as we first feared on Saturday."
Scott Shearer missed Tuesday's 3-2 loss to Middlesbrough with an ankle injury, though he will return to training soon, while David Worrall limped out of that game against Boro in the first half.
Raynor added: "It was just a bit of tightness. It was just precautionary, he just had a bit of a tight groin, but we don't want to take any chances even though he wanted to continue. We're hoping it's nothing too serious there."