Albanese initially declined to respond, saying he was recently married, but Osborne pressed, asking who he would choose if his marriage broke down, and Albanese replied, "Kylie, clearly."
"You'd marry Kylie, and shag her, and date her?" Osborne said.
"All of the above," Albanese replied.
Albanese was criticised by several opposition lawmakers, with senator Sarah Henderson saying the comments were "disrespectful to women, embarrassing to Australians and demean the office of Prime Minister."
"I apologise unequivocally for the comments," Albanese said in a statement on Monday.
A spokesperson for Minogue did not immediately respond to a request for comment.
Australia is famed for its informal culture, and politicians in the country are frequently quizzed during interviews on issues including pop culture and sport.