The couple, who are expecting a baby in the summer, revealed the news in The Times newspaper.
The notice, placed in the Forthcoming Marriages section, said: "Miss T. Rogers and Mr G. Osborne. The engagement is announced between Thea and George."
Mr Osborne, 49, revealed he was in a relationship with Ms Rogers, 39, last May.
Ms Rogers worked as the politician's Chief of Staff before becoming his special adviser. She is credited with helping him overhaul his image, including changing his hair and encouraging him to follow a healthier lifestyle.
Mr Osborne has two children with his ex-wife Frances, 52, who he announced his separation from in 2019. They are now divorced.
Mr Osborne was chancellor from 2010 to 2016 when David Cameron was prime minister. He served as the Conservative MP for Tatton from 2001 to 2017.
He left public office at the 2017 general election and went on to edit the Evening Standard newspaper until June 2020 when he became editor-in-chief.
He left the role in March and was recently made a partner at the investment bank Robey Warshaw.
Ms Rogers was awarded an OBE in Mr Cameron's resignation honours list and now works for Deliveroo as their chief customer officer.
She is expected to receive a multimillion pound windfall after the company launched on the London stock market last week.