Klopp signed the player for BVB in 2013, and he has blossomed into one of the deadliest attackers on the planet.
Liverpool have been looking for a striking sensation since the departure of Luis Suarez in 2014, and Aubameyang fits the profile of a world-class acquisition.
The player's record for his club is in the top echelon, scoring 120 goals in 189 appearances for Dortmund.
Klopp will be aiming to expand his squad in the coming weeks as the club looks forward to its return to the UEFA Champions League.
According to Olver, Liverpool are also "closing in on a deal" for Mohamed Salah of AS Roma, giving the Reds fresh options in attack.
The 25-year-old can play in a number of positions, but he would bring genuine width to Anfield.
Per Olver, Klopp continues to negotiate a price for the Eqyptian star after Roma turned down a £28 million offer from the Reds.
Football writer Sam McGuire is excited by Liverpool's potential attack next term:
Sam McGuire @SamMcGuire90
Naby and Coutinho in midfield. Salah and Mané on the flanks with Aubameyang and Firmino doing bits centrally Defending is overrated anyway https://t.co/JGYxsLuIji2017-6-18 19:17:07
Salah was a regular for his team last season, scoring 15 times and providing 11 assists in 29 starts in Serie A, according to WhoScored.com
The player signed for Chelsea in 2014 but failed to live up to expectations at Stamford Bridge in the challenging environment of English football.
This might force Klopp to think long and hard about completing a deal, but Salah has experienced a huge upturn in fortunes in Italy.
Aubameyang's signing would be a statement of intent from Liverpool as they attempt to take back their former place at the top table of European football.
A switch to the Reds would be a huge surprise, though, as the striker has been consistently linked with the biggest teams in the world, per the Telegraph.