Does God know who will be saved?

Yes.

This is different from predetermination, though. God knows everything that can happen, has happened and will happen. He has perfect foreknowledge - He knows who will accept Him and who will reject Him. Those He foreknew He predetermined to be conformed to the image of His Son.

God is omniscient (all knowing).

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Psalm 139:2-6 — You know when I sit down and when I rise up; You understand my thought from afar.
You scrutinize my path and my lying down, And are intimately acquainted with all my ways.
Even before there is a word on my tongue, Behold, O LORD, You know it all.
You have enclosed me behind and before, And laid Your hand upon me.
{Such} knowledge is too wonderful for me; It is {too} high, I cannot attain to it.

Isaiah 40:13-14 — Who has directed the Spirit of the LORD, Or as His counselor has informed Him?
With whom did He consult and {who} gave Him understanding? And {who} taught Him in the path of justice and taught Him knowledge And informed Him of the way of understanding?

Romans 8:29 — For those whom He foreknew, He also predestined {to become} conformed to the image of His Son, so that He would be the firstborn among many brethren;