At the same time, Mr. Bush has sought to inherit his brother’s support among neoconservatives. But his difficulty in measuring up to his brother in the eyes of those opinion-makers and donors was much in evidence over the past six weeks: In late March, his allies fielded angry calls from Sheldon Adelson, the casino billionaire who is a staunch supporter of Israel and gave tens of millions of dollars to “super PACs” in 2012, over a speech given by one of Mr. Bush’s foreign policy advisers, James A. Baker III. But Mr. Adelson gave the former president a lavish reception when he delivered the keynote address in late April to a gathering of the Republican Jewish Coalition, for which Mr. Adelson is a major benefactor.

Matthew Dowd, a strategist who worked on George W. Bush’s 2004 campaign, warned that Mr. Bush’s strong approval ratings do not translate into voters wanting to see his third term in office.

“And I think that’s the quandary that Jeb’s now got himself in. It’s more incumbent on him to say, ‘He’s my brother, I love my brother, but this is how my presidency would look differently,’ and show that.”

Mr. Bush seemed to be approaching his brother’s foreign policy record delicately from the first: In a major speech in Chicago in February, he articulated a recognizably muscular approach to global affairs, but still declared himself “my own man.”

Yet in recent weeks, as he strove to soothe supporters of Israel who questioned his commitment in closed-door meetings, Mr. Bush used his brother’s record on Israel as evidence of his own reliability — and went on to describe his brother as the person he listens to the most on Mideast issues.

It was a surprise to several of those who heard Mr. Bush say it — and to a number of people close to George W. Bush who learned of Mr. Bush’s remarks after the fact.