Coutts has netted just once this season, in the 2-1 loss to Walsall in the Football League Trophy earlier this month.
Fleck has yet to open his Blades account but goals run in the family genes.
His uncle, Robert Fleck, was a renowned striker at Rangers, Norwich City and Chelsea in the Eighties and Nineties.
Now Blades manager Chris Wilder is looking for goals from midfield to swell United’s threat going forward.
“Me and Paul had a chat with the manager last week and he said he wanted us to get some goals,” said 25-year-old Fleck, himself a former Rangers player.
“Sometimes, I look for a pass for a striker to score but that’s something I have got to improve and work on.
“It’s a good partnership with Couttsy. I knew him before I came here from Scotland and he’s a cracking player.
“You try and take something from everyone you play with. Everyone has got different attributes.”
Fleck has had to be patient for his chance at Bramall Lane, after arriving in the summer from Coventry City. But he got his chance, benefiting mainly from Chris Basham’s three-match ban for his red card at Scunthorpe United, and has cemented his midfield spot with some outstanding displays.
It is no coincidence with Fleck’s presence United have embarked on a nine-match unbeaten run in League One, which has seen them rocket up the table.
“We’re on a good run and had a good result Tuesday (a 3-0 win at Shrewsbury) but we’re experienced enough to know how quickly things can change.
“I had a little spell out which was disappointing but the team were doing well and so there were no complaints whatsoever.
“Obviously Bash’s suspension let me back in and I know I’ve got to battle for my place.
“When I was out, I was doing a little bit extra to keep on top of my fitness and making sure I was sharp in case I was called upon.”