The two nations were set to face each other four times in October and next February but they have been moved following talks
Australia and Bangladesh have agreed to move their upcoming T20 and Test series following talks between both cricket boards.
The Tigers were due to host the Aussies in two T20s in October, before a pair of Tests in February 2020.
But while the five-day games, which are part of the World Test Championship, have been shifted to June and July 2020, the white-ball contests will have to wait until 2021.
The dates for the limited-overs series, which will be increased to three matches, are yet to be confirmed but they will act as preparation for the World T20 in India later in the year.
"According to the FTP, we were supposed to host two Tests in February but now the Test series will be played in June-July 2020," said BCB cricket operations chairman Akram Khan.
"Initially we were expecting to host Australia for a two-match Twenty20 International series in October but now they have agreed to play three T20 internationals and it will be played ahead of the World T20 in India, though we are yet to finalise the date."
Australia began their WTC campaign by drawing the five-match Ashes series against England 2-2. Bangladesh begin their campaign against India in November.

David Warner's 166 settled the World Cup clash between the two nations in June
This is the second occasion in as many years that a scheduled series between the two countries has encountered problems.
In May 2018, Australia called off the visit of Bangladesh, who were due to play two Tests and three ODIs Down Under later that year.
The reason cited for the cancellation was lack of interest from local broadcasters, with the matches due to take place during the middle of the NRL and AFL seasons.
Cricket Australia James Sutherland insisted at the time that no repeat of the cancellation would occur due to the importance of the World Test Championship, which is currently in its first cycle.
"If we get drawn to play [Bangladesh], and we're drawn to play it at home, once you're in a championship situation, the context puts you in a position of wanting to win every game and needing the points and not compromising," he said.
"So you need to play at home and to try to win because those points count for something that at the end of the day might really matter to finishing one or two and qualifying for the final."