Central defender William Gallas, who was doubtful with a knee problem, will train later on Monday and the London club are hopeful he will also play, Redknapp told a news conference at Real's Bernabeu stadium.
"Yeah he's fine," he said of Bale, who has been in blistering form in Europe's elite club competition this season.
"He's going to train tonight and if he has no problems he'll play," he added. "Gallas will train. We're hoping he'll be fit."
However, midfielder Nico Kranjcar is struggling with a badly bruised foot sustained in Saturday's 0-0 draw at Wigan Athletic and Tottenham could be forced to field only six substitutes instead of the seven allowed by UEFA.
Wales international Bale has attracted the attention of a number of Europe's top clubs with his swashbuckling form, including a brilliant hat-trick in a 4-3 loss at holders Inter Milan in October.
His presence on Tuesday could be crucial for a Tottenham side making their debut in the Champions League and who have drawn their last three matches in all competitions 0-0.
"Gareth Bale gives us everything," Redknapp said. "He gives us goals and he gives us a fantastic pace and the ability to turn defence into attack very quickly."
Tottenham take on the nine-times winners three days after Real crashed to a shock home loss to Sporting Gijon in La Liga. The defeat spoiled their perfect home record in the league of 14 wins from 14 this season and ended coach Jose Mourinho's incredible nine-year unbeaten run in home league games.
"We often go home on a Saturday after losing at home and he hasn't done it for nine years," Redknapp said.
"So now he knows how the rest of us feel. Hopefully, it will be a bad week."
SPURS DREAM
Redknapp said his players would have to be at their best to have a chance of getting a positive result.
"I don't feel any pressure tomorrow. The only pressure is that we play well," he said.
"Nobody expected us to be here, let's be honest. We have to play well tomorrow. We're not just coming here to turn up.
"The dream is still there for us and we want to make that dream come true and go all the way.
"We feel we can give anyone a game. Maybe Real Madrid are favourites but we come here very confident and very optimistic we can do the job."
The winners of the two-legged quarter-final will play 2009 champions Barcelona or Shakhtar Donetsk of Ukraine in the semi-finals.
