The Bosnia and Herzegovina international has been restricted to a bit-part role at the Blades this season, due to the fine form of first-choice midfield three Ollie Norwood, John Fleck and John Lundstram.
But a combination of injury and fatigue for Lundstram has opened the door for Besic in recent weeks, and his latest start for the Blades came in Tuesday night's 1-0 defeat to Manchester City.
Blades fans have been impressed by the 27-year-old's mixture of bite and guile, and could get another chance to see him in action this weekend when United take on Millwall in the fourth round of the FA Cup at The Den.
Besic signed a new five-and-a-half year deal at Goodison Park in February 2016, but a succession of injuries curtailed his progress at his parent club.
And, speaking after the City game, Besic appeared to hint that he would be open to the possibility of exploring a prolonged stay at Bramall Lane under boss Chris Wilder.
"The people who need to know what I want, they know. So we will see what will happen," he said.
"It’s an unbelievable team, I’m really happy to be here and be part of this.
“I’m just really relaxed. It’s nothing big for me. As long as Chris Wilder is happy I will play for him.”