Bedevilled by persistent injury, King had hoped that his record of 27 games this season would convince the club that he was worth handing a fresh contract.
However, with King unable to train with the rest of the squad due to his knee complaint, Redknapp has been at pains to stress the need to live a healthy lifestyle if the club were to invest in him.
King's run in with the police in the early hours of Sunday morning in Soho in central London has made many at the club question if the player is heeding Redknapp's advice, according
to a report in the Independent newspaper.
Daniel Levy, the Spurs chairman whose pool King uses during the week to work on his knee, remains unconvinced of King's value if he fails to take proper care of himself.
King is likely to only be offered a pay-as-you-play deal with Tottenham though his options, despite the England international's exceptional talent, appear limited elsewhere.
The chances of him passing any medical at another club are negligible and he will now have to convince Tottenham that the weekend's misadventure is never to be repeated.
Meanwhile, Redknapp will introduce tough new rules to ban players from drinking alcohol during the football season.
'I just feel footballers shouldn't drink,' Redknapp said. 'I just think they should dedicate their lives to playing and I don't think there is any place in the game for drinking.
'Next year I'm definitely going to implement strong rules that drinking is a no-no. We'll talk to them, tell them where I see it all going.
'You wouldn't get these problems if the players weren't having a drink.'