During a news conference with reporters from seven Florida newspapers, Bush was asked if Jeb, also a Republican, should run for the top job.
"I would like to see Jeb run at some point in time, but I have no idea if that's his intention or not," said the president.
Bush said he has asked his brother about his plans, to no avail, the St. Petersburg Times reports.
The president was in Florida touting the new Mediare drug benefit.
During the 55-minute interview Bush reiterated his stance that oil drilling off the Florida coast should be 100 miles out.
That didn't eliminate any of the options Congress is looking at, including one plan opposed by the Florida delegation.
"I will support the nominee of the party," said Bush of Harris, who was Florida secretary of state and his Florida campaign coordinator for the 2000 election.
She has gotten weak support from the national Republican Party, which largely thinks Harris can't beat Nelson and is looking for a primary opponent to oppose her, the newspaper said.