Although commonly spoken, the phrase die to yourself, or dying to self, is not used explicitly in the Bible.
The Bible does seem to indicate pretty clearly that if you are a believer then you have already died and your life is now Christ's life. Or as Colossians 3 notes, your life is hidden with Christ in God:
3:1-4 Raised with Christ
If then you have been raised with Christ, seek the things that are above, where Christ is, seated at the right hand of God.
Set your minds on things that are above, not on things that are on earth.
For you have died, and your life is hidden with Christ in God.
When Christ who is your life appears, then you also will appear with him in glory.
There is certainly, especially in the Colossians 3 passage, the imperative to die to our flesh (our worldly desires), but it seems pretty clear that our "self", is no longer our flesh, but it is who we truly are in Christ. In other words as believers we are not to die to ourselves we have to encourage ourselves to be conformed to the image of Christ–which is precisely what God is working in us, see Philippians 1#1:6 in Colossians 3#3:10. Paul's argument is that our old person, our flesh, is already dead and gone, what we need to do is live fully in our new life which Christ has given us.
See also the New Bible commentary on page 1272 about Colossians 3.