Last month, Ms Duterte-Carpio said she would not run for president, a strategy used by her father who initially refused to run in 2015 before eventually joining the race later.
Broadcaster ABS-CBN interviewed Mr Duterte following Sunday’s surprise announcement that he would retire from politics.
"So is it clear, Sara-Go?,” a reporter asked, referring to Ms Duterte-Carpio and a senator, Christopher “Bong” Go, her potential running mate who could run for vice president.
"It is Sara-Go," Mr Duterte said in response. Mr Duterte then said he was not sure if she would run, because he does not discuss politics with his daughter, but this has not stopped widespread speculation in local media.
Ms Duterte-Carpio has also fuelled the rumours, setting up a Facebook page in July where she posted a video saying she wanted the public to “get to know her”.
Ms Duterte-Carpio has also taken the lead in public opinion polls on who is favoured to take the presidency.
Following her July debut on Facebook, "Run, Sara, run" posters and T-shirts were seen in cities across the country.
Like her father, Ms Duterte-Carpio trained as a lawyer and serves as mayor of Davao City, the Philippines’ third-largest metropolis, with a population of almost two million.
Mr Duterte was mayor of Davao for two decades before taking the presidency in 2016. She also carries her father's reputation for being tough and outspoken.
She once punched a court official who challenged her over illegal slum demolition. Ms Duterte-Carpio rides big motorcycles and her children are nicknamed Sharkie, Stingray and Stonefish.
Christopher Go served as special assistant to the president and is often described as Mr Duterte’s chief political aide. He filed his certificate of candidacy to join the race on Saturday.
But Ms Duterte-Carpio has been silent on whether she has changed her mind.
Although she has a strong lead in the opinion polls, Mr Go lacks a base of public support.
This is not for lack of effort – in 2018 he led the distribution of aid to fire-hit communities in Manila, leading some to accuse him of using government-supplied aid for political gain.
Analysts say it is crucial for Mr Duterte to have a loyal successor such as Ms Duterte-Carpio to insulate him from potential legal action – at home or by the International Criminal Court – over the thousands of state killings in his war on drugs since 2016.
Candidates have until Friday to register, but withdrawals and substitutions are allowed until November 15, leaving scope for last-minute changes of heart.