Can man’s will cooperate with God’s will in salvation?

Yes
God’s predestination is based on the decision we will make to accept His forgiveness.
John 1:12 But as many as received Him, to them He gave the right to become achildren of God, even bto those who believe in His name,[1]
We have the responsibility to believe in Christ because if we don’t we are then condemned by our lack of faith. Mark 16:16 says, “He that believeth and is baptized shall be saved; but he that believeth not shall be damned.” (kjv) We are also responsible to repent which must be our own decision, for what is the point of forced repentance? Luke 13:5 says, “I tell you, Nay: but, except ye repent, ye shall all likewise perish.” John 6:40 “For this is the will of My Father, that everyone who abeholds the Son and bbelieves in Him will have eternal life, and I Myself will craise him up on the last day.”[2] -“that everyone who believes in Him will have eternal life.” Is a conditional statement that must require a man to make the choice to believe before eternal Life may be given.
No
If God’s predestination or salvation is based off of a decision of man then His decision would be reactionary and this is against His nature. John 1:13 says, "Which were born, not of blood, nor of the will of the flesh, nor of the will of man, but of God."
“It is God which worketh in you both to will and to do of his good pleasure” Phi 2:13.
Romans 9:16 says, "So then it does not depend on the man who wills or the man who aruns, but on bGod who has mercy.[3]"
The result of grace working in one’s soul caused the repentant sinner to have a new affection for God that now desires God more than he desires sin. Repentance is not something that the sinner is adding in addition to faith as a work, but both repentance and faith are seen as the infallible result of the new birth that is applied to sinners by the Holy Spirit
John 6:39“This is the will of Him who sent Me, that of aall that He has given Me I blose nothing, but craise it up on the last day[4]
God has given or “chosen” those to believe in Him. All those who He has chosen to believe shall have eternal life.

a John 11:52; Gal 3:26
b John 1:7; 3:18; 1 John 3:23; 5:13
[1]New American Standard Bible : 1995 update. 1995 . The Lockman Foundation: LaHabra, CA
a John 12:45; 14:17, 19
b John 3:16
c Matt 10:15; John 6:39, 44, 54; 11:24
[2]New American Standard Bible : 1995 update. 1995 . The Lockman Foundation: LaHabra, CA
a Gal 2:2
b Eph 2:8
[3]New American Standard Bible : 1995 update. 1995 . The Lockman Foundation: LaHabra, CA
a John 6:37; 17:2, 24
b John 17:12; 18:9
c Matt 10:15; John 6:40, 44, 54; 11:24
[4]New American Standard Bible : 1995 update. 1995 . The Lockman Foundation: LaHabra, CA