Lying

I have 13 year old twin boys who struggle with integrity and lying.
When my husband and I have confronted them in situations when we know they are lying, they hold fast to the lie and try to move on from it. How do we go about correcting the behavior when they insist they aren't lying, but we have evidence from another family member that what they are saying isn't true?
We feel there is a trust and attachment issue with our son's. They are very attached to each other but not as attached to us. How do we strengthen their attachment to us so they will trust us and feel safe to tell us the truth?