There are basically two theological approaches for interpreting this passage:

(1) Calvinistic Approach: Apart from a Divine irresistible work of regeneration (by which God changes a chosen individual’s nature and desires), mankind cannot willingly seek to know, understand, or follow God.

(2) Non-Calvinistic (Provisionist) Approach: Apart from God’s gracious initiative in bringing His Son, the Holy Spirit, and the inspired gospel appeal, no one can merit salvation or consistently seek to obey God in a way that will attain his own righteousness

Paul is not teaching that no one can believe in God. He is teaching that no one can perfectly fulfill the commands of the law.

His main point is contending that the Jews are thinking to highly of themselves, and they they also are under sin and this passage in Romans is Paul’s way of demonstrating that both Jews and Gentiles are under the power of sin, using the Jewish scriptures to make his point clear.

Here is the full breakdown of the OT references in his quote:

Romans 3:11: "there is no one who understands; there is no one who seeks God."

Reference: Psalm 14:2; Psalm 53:2
Romans 3:12: "All have turned away, they have together become worthless; there is no one who does good, not even one."

Reference: Psalm 14:3; Psalm 53:3
Romans 3:13: "Their throats are open graves; their tongues practice deceit."

Reference: Psalm 5:9
"The poison of vipers is on their lips."
Reference: Psalm 140:3
Romans 3:14: "Their mouths are full of cursing and bitterness."

Reference: Psalm 10:7
Romans 3:15: "Their feet are swift to shed blood;"

Reference: Isaiah 59:7
Romans 3:16: "ruin and misery mark their ways,"

Reference: Isaiah 59:7
Romans 3:17: "and the way of peace they do not know."

Reference: Isaiah 59:8
Romans 3:18: "There is no fear of God before their eyes."

Reference: Psalm 36:1

The main point Paul is making is not about whether man can seek after God or Jesus, but that none of us would be able to seek out true righteousness on our own accord.

Just because no man seeks God, does not mean that man can not respond when God reaches out to us?

Paul is essentially arguing that the whole concept of who is and who is not righteous is meaningless - faith is what matters. Nothing we can do, or not do, can save us, we cannot save ourselves.

Thankfully, God has given us the gift of the Gospel and is drawing all mankind to himself(John 12:32) and is commanding everyone everywhere to repent(Acts 17:30-31). The gospel message is sufficient as the means for God to reach out to mankind. God does not need to regenerate man first with a special grace sprinkling before they are able to respond positively to the Gospel.

Calvinists read this text to mean that our lack of initiative somehow proves our inability to respond positively to His initiative. ~ Leighton Flowers

Paul is teaching that no one can perfectly fulfill the commands of the law - Paul is not teaching, as the Calvinists would presume, that no one can believe in God.

Humanity's inability to be perfect does not mean that humanity is unable to confess our imperfections to God.

When looking at Pauls original citation of Psalm 14:2-3, it is contrasting the unbelieving fool against the righteous who wait on God. The unbelieving fool is the one who turned astray. That passage is not teaching Total Depravity but is rather teaching that no one can pursue or attain righteousness by works of the law.

This passage cannot mean what the Calvists think it means. The passage says that none is righteous no not one but then in the next passage - God calls Abraham righteous. Paul is obviously not contradicting himself but is instead talking about two different types of righteousness and seeking. No one is righteous or seeking through works of the law, so instead we must seek righteousness through faith in Christ.