They claim that Bush wants to run but will test the waters to gauge the reaction to his ideas, such as immigration reform and support for the controversial Common Core education standards, and his last name.
He is also wants to see reception of his wife and family, who they said are not keen on him running.
Bush has said he would try to make his mind up by the end of the year, but has now become more serious about testing his ideas in public.
Bush is expecting the right wing of the GOP to be his biggest challenge because of his more moderate positions on immigration and education and is eager to discover if he can win some over before making his mind up.
He polls well in primary states, and that is encouraging supporters, including some in the conservative, Tea Party and evangelical movements who know him well, as well as establishment Republicans who see him as an alternative to New Jersey Gov. Chris Christie.
His brother pushed Jeb Bush's case on CNN last week. “I hope Jeb runs,” George W. Bush said. “I think he would be a great president. I have no clue what's on his mind and we will talk when he's ready. I noticed he's moving around the country quite a bit.”
Paul Bedard, the Washington Examiner's "Washington Secrets" columnist, can be contacted at firstname.lastname@example.org.