Vicente is leaving Valencia as his contract expires next month.
"They have twice tried to sign me. The first time when I was only 16. The second time was after we had just won La Liga and the UEFA Cup with Valencia," Vicente told Cadena Ser.
"I was at Euro 2004 in Portugal with the national side when [Real Madrid president] Florentino [Perez] told me he wanted to sign me. He did not convince me as I just wanted to renew with Valencia and the president said no.