The club are no doubt working behind the scenes to tie him down for the next few years. Indeed, they may even have the option to extend his current deal by another year, as has been the case with several players over the last few years.
According to Sky Sports, the right-back signed a contract extension only a year ago.
Regardless, Celtic must offer Fisher a new, longer deal for three reasons: form, potential and security.
The 19-year-old has made 11 first-team appearances so far this season. Six of those have come as part of a defence which helped Fraser Forster go 11 consecutive league matches without conceding—breaking Charlie Shaw's 100-year-old clean sheet record in the process.
Despite being young and inexperienced, the youngster has comfortably slotted in alongside Virgil van Dijk, Efe Ambrose and Emilio Izaguirre in a back four which has conceded just 12 league goals all season.
The defender is also the first graduate direct from the club's youth system to manage a regular run in the senior side since the emergence of winger James Forrest in 2010.
Forrest's progress may have stalled in recent seasons, but when on form, he can still be almost unplayable.
Right-backs are also a different breed than wingers. Where Forrest was expected to entertain and create, Fisher need only stay consistent and show a willingness to join in attacks.
If current form were not reason enough to offer him a new deal, then Fisher's potential for further improvement surely would be.
For all his impressive performances, there is still substantial scope for enhancement. His defensive positioning can be suspect at times, especially when it comes to his tendency to allow opposition wingers the opportunity to slip inside him and run at the centre-backs.
He also appears hesitant to fully commit to attacks at times—something generally expected of Celtic full-backs.
The good news for the Hoops is that these supposed weaknesses are things that can be improved with more games in the first team and the confidence and experience that carries with it.
It helps that legendary right-back Danny McGrain trains the club's youngsters.
The ex-Celt—who was regarded as a world-class full-back during the 1960s, '70s and '80s—told the Daily Record back in November 2013 how highly he rates Fisher:
I like what I've seen of Darnell so far. First and foremost at Celtic you need to be able to defend—then you need to be a good counter-attacker, an overlapper who can cross.
He's already beginning to build a reputation of being pacy and hard in the tackle.
But he still has a lot to learn and he knows that.
The Daily Record notes that England Under-21 manager Gareth Southgate is another admirer, increasing speculation that Fisher is close to a call-up despite his relative inexperience at the club level.
Even the biggest obstacles to Fisher's continued involvement at the senior level could work for him when it comes to winning a new deal.
Injuries to Adam Matthews and Mikael Lustig allowed him this extended run in the first team. Both suffer from recurring injury problems—Matthews has missed large patches of his Celtic career due to injuries of varying degrees of severity. Lustig, meanwhile, has been plagued by hip injuries since shortly after he joined and is not due back until April from his latest layoff.
Nonetheless, the misfortune of his team-mates is not the upshot for Fisher, though he did admit in an interview with The Scotsman that he fully intended to take advantage of the experience that afforded him:
You have to try and express yourself when you get the change. Obvsiously Mika [Mikael Lustig] will be out for a few months now but I wouldn't say I was happy when I heard that.
I suppose it's a bonus for me as there might be more chances and I have to take any that come my way.
Both Matthews and Lustig are highly rated individuals who have been attracting interest from a variety of clubs and perform consistently well in the Champions League.
It would not be surprising if either, or both, decided to move on in the next year or so.
With that in mind, securing Fisher—a player who has shown himself capable of filling in for two of the side's best players—to a long-term contract makes sense for the club, even if manager Neil Lennon's overarching intention would be to sign a "ready-made" replacement in that scenario.
Given his solid performances and the trend of rewarding a player in fine form with renewed personal terms and an extended deal—not to mention the fact he is knocking on the door of an England Under-21 call-up and his main opponents for a regular starting spot are attracting widespread attention—it would not be surprising to see a club statement released in the next few weeks announcing that Fisher has been handed a new contract.
It almost seems inevitable.