"I think it's more than likely that he's (Jeb) giving this a serious thought in moving forward," George P Bush told ABC News.
Asked if it was "more than likely" that his father will run for presidency as the Republican candidate, George said, "If you had asked me a few years back, I would've said it was less likely. The family will be behind him 100 per cent if he decides to do it."
A former Republican governor of Florida, 61-year-old Jeb is a favourite of the party's establishment wing. If he runs, he is expected to draw on a deep well of support among the donor network cultivated by his father and brother.
Jeb has said he will only launch a bid if he can campaign unapologetically on the issues that are important to him. He has said he is likely to decide on a run after November's midterm elections, by the end of the year.
More From This Section
The position carries a portfolio ranging from managing the state's land and mineral resources to administering programmes for veterans.
It also oversees the Permanent School Fund, which is the nation's largest educational endowment at USD 37.7 billion.
Though this is his first run for office, George was immersed in Republican politics at a young age.
When he was 12-years-old, he recited the pledge of allegiance at the 1988 Republican National Convention, which nominated his grandfather, George H W Bush. In his 20s, he campaigned for his uncle George W Bush's presidential run.
"But I said from day one of my campaign, 23 months ago, that I am a man of my own right, who stands on my own two feet with my vision. And I need folks to evaluate me based on what I bring to the table," the 38-year-old said.
If George wins next Tuesday, he will hold the distinction of being the only Bush to win their first campaign.