No, there is no such reference in the Qur'an/Hadith, to my knowledge.
The very idea of being "godlike" is problematic from an Islamic standpoint. The Qur'an contains various verses that make it clear that God is unlike any of His creation.
Qur'an 42:11 فاطر السماوات والأرض جعل لكم من أنفسكم أزواجا ومن الأنعام أزواجا يذرؤكم فيه ليس كمثله شيء وهو السميع البصير ([He is] Creator of the heavens and the earth. He has made for you from yourselves, mates, and among the cattle, mates; He multiplies you thereby. There is nothing like unto Him, and He is the Hearing, the Seeing)
Moreover, Allah (ﷻ) even dismisses the notion of "sons/children of God":
Qur'an 5:18 وقالت اليهود والنصارى نحن أبناء الله وأحباؤه قل فلم يعذبكم بذنوبكم بل أنتم بشر ممن خلق يغفر لمن يشاء ويعذب من يشاء ولله ملك السماوات والأرض وما بينهما وإليه المصير (And (both) the Jews and the Christians say: "We are the children of Allah and His loved ones." Say: "Why then does He punish you for your sins?" Nay, you are but human beings, of those He has created, He forgives whom He wills and He punishes whom He wills. And to Allah belongs the dominion of the heavens and the earth and all that is between them, and to Him is the return (of all))
Islam is very clear that humans are just creations of God, nothing more. Even the best of humanity and leader of the Prophets, Prophet Muhammad (ﷺ), has never been attributed with anything other than being a [non-divine] regular man. Allah (ﷻ) commanded him to say and affirm to the people:
Quran 18:110 قل إنما أنا بشر مثلكم يوحى إلي أنما إلهكم إله واحد فمن كان يرجو لقاء ربه فليعمل عملا صالحا ولا يشرك بعبادة ربه أحدا (Say, "I am only a man like you, to whom has been revealed that your god is one God. So whoever would hope for the meeting with his Lord - let him do righteous work and not associate in the worship of his Lord anyone.")
On the other hand, what you describe of "connecting/knowing/being intimate with The Supreme One/Most High GOD" is definitely possible. Prophet Abraham (peace be upon him) had the title of Khalil-Allah (Friend of God).
Qur'an 4:125 ومن أحسن دينا ممن أسلم وجهه لله وهو محسن واتبع ملة إبراهيم حنيفا واتخذ الله إبراهيم خليلا And who is better in religion than one who submits himself to Allah while being a doer of good and follows the religion of Abraham, inclining toward truth? And Allah took Abraham as an intimate friend.
Islam teaches us that we can come closer and closer to Allah (ﷻ) until we become His 'Wali' which is a pious worshiper who is his close friend. Anyone has potential to reach this level through piety and righteous deeds.
Qur'an 10:62 ألا إن أولياء الله لا خوف عليهم ولا هم يحزنون (Behold! verily on the friends of Allah there is no fear, nor shall they grieve)
In conclusion, there is no concept of "divine humans" nor any concept of being "children of God" but there is a concept of being close/intimate Friends of God (Awliya).
Hope this helps.