"Jeb Bush has not ruled out running. He does intend to give it thoughtful consideration in the coming weeks and months," said his spokeswoman, Kristy Campbell. Bush is also the son of former President George H.W. Bush.
Martinez, who was narrowly elected to his first term from Florida in 2004, announced Tuesday he would not run for a second term because he wanted to spend more time with his family and friends.
The former governor "is weighing a variety of factors" and does not expect to make a decision before the end of the year about a potential candidacy, Campbell said.
She declined to elaborate on those factors.
Martinez was considered vulnerable in part because of his strong ties to the current president, who chose Martinez as secretary for housing and urban development during his first term.
Jeb Bush, 55, served two terms as governor from 1999 to 2007 and was consistently more popular in Florida than his brother, whose approval rating dropped to an all-time low for a U.S. president last month when 76 percent of respondents said they disapproved of his performance.
Florida backed Democratic President-elect Barack Obama in the November election and its other U.S. senator, Bill Nelson, is a Democrat. Republicans control both houses in the state Legislature and the current governor, Charlie Crist, is a Republican.
