Romans 16:20 is likely influenced/shaped by Genesis 3:15 and hearkens back to that passage:
From Henry Alford's GNT:
συντρ. τ. σατ. is a similitude from Genesis 3:15.
συντρίψει, not as Stuart, ‘for optative,’ nor does it express any wish, but a prophetic assurance and encouragement in bearing up against all adversaries, that it would not be long before the great Adversary himself would be bruised under their feet.
From Ellicott's Commentary for English Readers:
"Bruise.—With reference to Genesis 3:15."
From the Cambridge Bible for Schools and Colleges:
"shall bruise Satan, &c.] The very first promise of Redemption (Genesis 3:15,) is doubtless here referred to.—The 'Enemy who soweth tares' had been already 'bruised' by the Redeemer, in His triumphant work; and that victory would be, in due time, realized in the personal ('under your feet,') triumph over sin and death, and final deliverance from all trial, of each of His followers."
From John Gill's Exposition of the Bible:
"so here is a manifest allusion to what was said by way of threatening to him, 'it', the woman's seed, 'shall bruise thy head', ( Genesis 3:15 ); and which has had its accomplishment in Christ, who has not only destroyed the works of the devil, but him himself, and spoiled his principalities and powers, and bruised him and them under his feet, when he led captivity captive"
Barnes also writes:
"Will bruise - The 'language' here refers to the prediction in Genesis 3:15. It here means to 'subdue, to gain the victory over.' It denotes Paul's confidence that they 'would' gain the victory, and would be able to overcome all the arts of those who were endeavoring to sow discord and contention among them."
More recent commentators have also pointed to the "clear" Genesis allusion in Romans 16:20. See Tremper Longman III and Daniel G. Reid, "God Is A Warrior."