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.