The German champions are said to have earmarked the Scotland international as their number one choice to replace Alphonso Davies, who has apparently verbally agreed a move to Real Madrid.
Bayern - who will part ways with Thomas Tuchel in the summer - are hopeful that Robertson will feel that now is the time to embark on a new challenge after seven successful years at Anfield, especially with Jurgen Klopp also moving on.
However, the 29-year-old's contract with the Premier League leaders still has just over two years left to run, and despite a serious shoulder injury earlier this season, he remains an integral part of Klopp's setup.
As a result, Football Insider claims that Liverpool expect to rebuff interest from Bayern for Robertson this summer and are confident that they will continue working with the defender for the long-term.
The Reds supposedly have no fears over Robertson wanting to make the move to the Allianz Arena, and they do not believe that Klopp's impending departure will have any impact on his future.
A former Queen's Park and Dundee United youngster, Robertson arrived at Anfield from Hull City in the summer of 2017, since when he has come up with nine goals and 64 assists in 282 matches for Liverpool.
The 29-year-old has set up 58 goals in the Premier League, the joint-most of any defender in the history of the competition - only teammate and fellow full-back Trent Alexander-Arnold has provided as many assists.
Robertson has won seven different pieces of silverware since signing for the Reds, clinching two EFL Cups in addition to the Champions League, Premier League, FA Cup, Club World Cup, UEFA Super Cup and Community Shield.
However, the Scotsman has just one goal and one assist under his belt from 15 matches in 2023-24, having missed three months of action after dislocating his shoulder while playing for Scotland.
Robertson also sat out Wednesday's 3-0 FA Cup fifth-round success over Southampton due to illness, and Klopp could not guarantee that he would be fit for the weekend's Premier League encounter with Nottingham Forest.
While Robertson is expected to stay put at Anfield, Liverpool stand to lose Joel Matip at the end of the season when his contract expires, although Klopp suggested that the Cameroonian could be offered a new deal after his season-ending ACL injury.
The likes of Rhys Williams, Nat Phillips and Sepp van den Berg could depart permanently, though, and the Reds have supposedly made defensive signings a priority for the upcoming summer window.
Liverpool hold a one-point lead atop the table ahead of the trip to Nottingham Forest's City Ground, where they have failed to win any of their previous six Premier League fixtures.