According to Bild's Roland Palmert (h/t the Mirror's Darren Wells), Jovic has caught Klopp's attention after a superb start to the season, which has seen him bag 12 goals in 15 appearances in all competitions and could try to land him in January.
However, he has caught the eye of Europe's other elite sides, too, and Frankfurt board member Fredi Bobic said: "Our grandstand has been full of scouts from the absolute top clubs for weeks. They send their best people. I'll stay calm, that's also recognition for us. We will be noticed again in Europe."
Jovic has only started nine matches, but he has been on fire in front of goal, averaging a goal every 70 minutes.
The 20-year-old's numbers have been improved by a stunning five-goal haul he achieved in a 7-1 win over Fortuna Dusseldorf in October:
Nevertheless, he has backed that up with a further seven goals and three assists across his 14 other appearances this season.
The youngster is having a breakout year for Frankfurt, having also netted seven of his nine goals for them last season in 2018.
On this form, the Bundesliga starlet could make a fine back-up at Anfield, but according to Goal's Neil Jones the Reds are set to stick with their current cover options:
What's more, the Serb is actually on loan at Frankfurt from Benfica, which would likely preclude him from going anywhere in January.
Per football journalist Chris Williams, it seems his stay in Germany will then be made permanent:
Of course, there would be nothing stopping Frankfurt from immediately selling him on for a hefty profit, should Barcelona, Bayern or Liverpool come in for him.
For now, though, he'd likely be best served by remaining where he is, as he'll enjoy much more first-team football in Frankfurt than he would at one of the elite sides.