The 24-year-old is a leading transfer target for the Reds this summer and the Germany international is understood to have a release clause of £53 million written into his contract.
Werner had already struck 27 goals in all competitions for the Bundesliga outfit before this game, despite failing to find the net in their 1-1 draw with Freiburg as the German league returned last weekend.
But the striker quickly shook off that ring rust with a magnificent all-round display against struggling Mainz on an afternoon where he also became the club's all-time leading goal scorer.
As Julien Nagelsmann's side dominated the opening proceedings, Werner was looking a threat in a three-man attack with Youssef Poulsen and Marcel Sabitzer.
It took him just 11 minutes to get off the mark. Konrad Laimer burst down the right, evaded two challenges and found Werner inside the box with a low cross and the striker - with an instinctive finish - calmly diverted it into the far corner.
It highlighted just how potent Werner can be in front of goal in a central position, even if he is deployed as a wide forward in Leipzig's 3-4-3 formation.
What was also evident was his intelligent link-up play with Poulsen and Sabitzer - a clear indication of his suitability to fit into a similar system such as Liverpool's.
Poulsen made it 2-0 after heading home from Sabitzer's clipped cross and shortly after, Werner very nearly made it three with a thumping header - only for Mainz goalkeeper Muller to tip it around the post.
The German was proving to be a thorn in the side to the Mainz backline with his running in behind and exchanging neat one-twos on the edge of the box. He played his part in the third, digging out a shot for Muller to parry before Sabitzer finished confidently into the top corner.
Werner showed no signs of letting up in his energy or hunger to add to his goal tally and got his second shortly after the break, tapping home from a superb team move made by Kevin Kampl. There again, onlooking scouts would have identified his attacking movement, waiting patiently in the six-yard box to slot home after sneaking through the middle.
With Leipzig cruising towards another thumping victory over Mainz, it seemed only a matter of time before they would grab a fifth and Werner rounded off a fine afternoon a marvellous hat-trick. Nordi Mukiele launched a 50-yard ball over the top and Werner used his blistering pace to get ahead of his marker before dinking past the onrushing Muller.
Chelsea and Bayern Munich have also been credited with an interest but it is believed the forward favours a move to Merseyside to link up with Reds boss Jurgen Klopp. Should they fail to get a deal across the line, Liverpool's loss will be another team's gain.