Lions coach Warren Gatland expressed his anger at the treatment received by his scrum-half during the All Blacks' 30-15 victory at the weekend.
Kaino was one player highlighted for focusing on Murray after a lunging tackle on the Ireland international's standing leg as he kicked from the back of the scrum.
New Zealand coach Steve Hansen was quick to defend his players on Monday and Kaino has reiterated there was no specific plan to take out Murray.
"No, it's never our intent to go out and injure someone," Kaino said ahead of Saturday's second Test.
"I guess people have their opinions on it and all I can say is it wasn't my intention to go out there and target his planted foot.
"It's never nice when you have things done to you outside the laws. The way we do things is within the spirit of the game."
When asked about the specific challenge, he added: "I didn't go in to tackle him. I rolled into his leg. What I was trying to do was... his swinging foot, if you can disrupt that it's like an ankle tap so you can disrupt the kick.
"My timing was off and I rolled into his planted foot."