Bale has been ruled out of Saturday's Euro 2012 qualifier against England with a hamstring injury, and Speed said on Thursday that Bale had picked up the problem at Spurs.
However, Redknapp is adamant the winger was feeling fine after completing 90 minutes for Tottenham in their goalless draw against West Ham at the weekend.
"I wouldn't have let him go if he had an injury," Redknapp told the Daily Mail. "Gareth played for 90 minutes against West Ham on Saturday and they tell me the figures from that game show he covered more ground than he's ever done before.
"There was no sign of a hamstring injury. He's not come in after the game and said his hamstrings were tight. When he left here on Saturday, off he went and he certainly wasn't injured.
"He hasn't had a hamstring injury. He had a back, then he had a tight thigh and we've been very careful with him. Now he's got a hamstring. He must've done it with Wales. I'm not blaming anyone. It's no one's fault."
With Tottenham's Champions League quarter-final against Real Madrid in less than a fortnight, Redknapp is hopeful Bale will be ready after a week's rest, although he admits there is a risk of further aggravating the injury.
"The first I heard of it was at 10 o'clock on Wednesday night when our physio called me," Redknapp said. "I feared the worst but he's seen our physio and it's only tightness. There's no tear but if he plays and it pings, he's had it."