ANSWER: We would not have a Bible if God was unwilling to communicate with sinners! Scripture was inspired and written specifically so that all humans would have the means to come to repentance and change their ways. Scripture is full of examples where God was not only willing to hear sinners but also, in some cases, to initiate contact and talk with them!
God initiated a discussion with both Adam and Eve AFTER they became sinners by eating of the tree of good and evil (Genesis 3:8 - 19). He also initiated a conversation with Cain after his offering was rejected (Genesis 4:6 - 7) and after he murdered, in cold blood, his younger brother Abel (verses 9 to 15). The Lord also spoke directly to Eliphaz, one of Job's three friends, in order to reprimand him for speaking lies during Job's trials (Job 42:7 - 8). King Hezekiah of Israel prayed that God would heal him of his deadly illness. His petition, given in humility and tears, was heard and he was healed (2Kings 20:1 - 6).
Jesus, as God in the flesh, spent his entire ministry preaching the gospel and calling sinners to repentance (Luke 5:32). He interacted, on a regular basis, with those considered the most disobedient (e.g. tax collectors and harlots, see Matthew 9:10 - 11, 11:19, 21:32, Mark 2:16, Luke 5:30, etc.). In fact, apart from John the Baptist, Christ NEVER spoke to a converted human, meaning someone who had the Holy Spirit in them, his entire life! This includes both his mother and the disciples!
The truth is, we are all sinners who have fallen far short of his glory (Romans 3:23). Our heavenly Father is more than willing to look upon and respond to those willing to humble themselves before him (Isaiah 66:2). The cries of the poor do reach his ears (Job 34:28).
The Lord has stated, however, that he will not respond to those who are indulging in evil and refuse to repent (Proverbs 1:24 - 28, 28:9, Daniel 9:13 - 14, 1Peter 3:12). In the Old Testament, He warned ancient Israel of the consequences of rejecting him as their King (1Samuel 8:11 - 17). He then stated, because of their refusal to repent, that when they cry out to him for relief from the king they demanded that he would not answer them (1Samuel 8:18).
God does hear the prayers of sinners and then decides, based on his wisdom and love, how he will answer them. After all, since the time of Adam, all the prayers that have been directed toward him have come from imperfect humans.