Both players missed the trip to Coventry City at the weekend, Fleck struggling with a groin complaint and Liverpool loanee Davies damaging his calf.
United were initially hopeful that the pair would be fit to face the Seasiders, but Heckingbottom revealed today that that would “probably not” be the case.
“It’s easier to go through who we have got, isn’t it?” the United manager added.
“We’ve got over 50 per cent of our outfield first team out injured. We’re hoping [they’ll be back for the weekend]. They’re not far away.
“Flecky didn’t train with us today; the plan was to get him in and around it tomorrow. That was the plan. Because of how light we are, with a muscle injury, I just don’t see it as the right risk.
“We can’t afford players out. You can see the bench, we have to be a little bit cautious.”
Oli McBurnie, who came off the bench at Coventry and was substituted again after picking up a dead leg, will be given “every chance”, but Charlie Goode and George Baldock will both miss out as the Blades look to bounce back from their 4-1 drubbing at Coventry.
“Oli’s really struggling today,” Heckingbottom added.
“He was desperate not to come off but he couldn’t get about the pitch [at Coventry]. To protect him down the line he had to come off.
“It can sometimes settle down in 48 hours so we’ll give him every chance. We’re not ruling him out.”
Goode has returned to Brentford for treatment on a cartilage problem.
“He’ll be out for a couple of weeks,” Heckingbottom added.
“He’s not our player, we can’t take risks with him because Brentford are involved. It’s flared up, he’s been to see a specialist and he’s getting a clean-out injection which will take him to after the international break.
“He’s back with Brentford in the short term and then he’ll be back with us.”