Tudor has been named new manager after the dismissal of Thomas Frank last week. Amid the Dane's departure, Romero sparked controversy by criticising the board publicly before being sent off in defeat at Manchester United.
Writing for The Sun, Redknapp stated: "There are still a few players Tudor will need to deal with and none more so than the captain Cristian Romero.
"You cannot take the armband away from him, as much as that would seem an easy fix. Strip Romero of the captaincy and then you have lost him from day one.
"For the short term, you need him on your side and with you — but there needs to be a frank conversation about his lack of discipline.
"We can all see there is quality there, but Tudor has to haul him into his office and tell him how much he is needed out on the pitch. You have to be clear.
"The Argentine centre-back will know his reputation is disappearing down the drain with every diabolical or rash challenge that ends up with another suspension. He is the only one who can change that and the manager needs to support him to do so.
"A fine example for Romero and the rest of the squad to follow is 19-year-old midfielder Archie Gray.
"He was Tottenham’s player of the year for me last season and delivers whenever or wherever you play him.
"Gray is a future Tottenham captain and everyone should be looking to him.
"He puts on a performance and shows commitment and desire whether at right-back, centre-back or in midfield.
"The kids at Tottenham, both Gray and Lucas Bergvall, have been showing up the leaders for far too long and it is time the more senior figures followed their lead."