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Romans 8
3For what the Law could not do, weak as it was through the flesh, God did: sending His own Son in the [b]likeness [/b]
of sinful flesh and as an offering for sin, 
Paul has given us a good definition of the sinful nature in Rom 3 – showing that in that regard the sinful nature has nothing good in it – it is “full of sin” it is our sinful inclination to rebel against God’s Word.
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Rom 3 
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The Sinful nature of all mankind –
9 What then? Are we better than they? Not at all; for we have already charged that both Jews and Greeks are all under sin;
10 as it is written, "" THERE IS NONE RIGHTEOUS, NOT EVEN ONE;
11 THERE IS NONE WHO UNDERSTANDS, THERE IS NONE WHO SEEKS FOR GOD;
12 ALL HAVE TURNED ASIDE, TOGETHER THEY HAVE BECOME USELESS; THERE IS NONE WHO DOES GOOD, THERE IS NOT EVEN ONE.''
13 "" THEIR THROAT IS AN OPEN GRAVE, WITH THEIR TONGUES THEY KEEP DECEIVING,'' "" THE POISON OF ASPS IS UNDER THEIR LIPS'';
14 "" WHOSE MOUTH IS FULL OF CURSING AND BITTERNESS'';
15 "" THEIR FEET ARE SWIFT TO SHED BLOOD,
16 DESTRUCTION AND MISERY ARE IN THEIR PATHS,
17 AND THE PATH OF PEACE THEY HAVE NOT KNOWN.''
18 "" THERE IS NO FEAR OF GOD BEFORE THEIR EYES.''
19 Now we know that whatever the Law says, it speaks to those who are under the Law, so that every mouth may be closed and all the world may become accountable to God;
20 because by the works of the Law no flesh will be justified in His sight; for through the Law comes the knowledge of sin.
By contrast it is observed that "by the works of the LAW CHRIST WAS justified even in His human sinless flesh".
By contrast – only depravity “Dwells in us” by nature. We are “by nature the children of wrath” Eph 2:3-6.
And as Paul points out in Rom 7 – IN him is only depravity apart from the New Creation. (Good point LeBuick)
Rom 7
17 Now then it is no more I that do it, but sin that dwelleth in me. 
18 For I know that in me (that is, in my flesh,) dwelleth no good thing: for to will is present with me; but how to perform that which is good I find not. 
19 For the good that I would I do not: but the evil which I would not, that I do. 
20 Now if I do that I would not, it is no more I that do it, but sin that dwelleth in me. 21 I find then a law, that, when I would do good, evil is present with me.
In the text above however we do have a good argument for Christ also having a “sinful” nature. But notice that the text says “in the likeness of our sinful flesh”. When Paul uses that term – he shows in several cases that he does not mean to imply an exact equivalence.
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Notice Eph 4
22that, in reference to your former manner of life, you lay aside the old self, which is being corrupted in accordance with the lusts of deceit, 
23and that you be renewed in the spirit of your mind, 
24and put on the new self, which in the likeness of God has been created in righteousness and holiness of the truth. 
So while an argument could be made that “Sinful flesh” is to include the idea of sinful nature seen in Rom 3 – and that Rom 8 is making an exact equivalence claiming that Christ’s nature was totally depraved and “sinFUL” just as ours is – the example from Eph 4 above shows that this is not likely. For it is agreed that we are not in fact “God”. In the Likeness of God – does not make us anything like God in terms of exact equivalence. Christ took upon Himself our fallen – weakened human nature as opposed to the unfallen – perfect – flawless human nature of Adam. But that is as far as it went.