Perth Now is reporting that Ms Bishop has decided it “is in the interests of the nation” that she retire as a minister.
Fairfax News says she’s expected to announce her decision on Sunday.
Ms Bishop, 62, was deputy Liberal leader for more than a decade but she only received 11 votes in last Friday’s leadership ballot after Malcolm Turnbull called a spill.
The long-serving foreign minister returned to Perth on Saturday.
“I am going to consider all my options and I am going to focus on running (in the City to Surf),” she told the Sunday Times.
“Once I have considered my options I will make a statement.”
Education Minister Simon Birmingham earlier on Sunday said Ms Bishop was “the most significant woman in the history of the Liberal Party” and deserved to make a decision about her future without external pressure.
“We would love to see Julie continue, but that really is up to Julie,” he told the ABC on Sunday.
New prime minister Scott Morrison, who won the leadership ballot against Peter Dutton, is weighing up his new ministry to reunite the Liberal party.