No, no choice. Not only can the unsaved choose to reject God's gospel but do so consistently without exception. They do because they are in the flesh, spiritually blind, trusting in their works for salvation. It is God who at a certain point, takes His elect from spiritual death unto spiritual life. Those He makes spiritually alive He gives a new spirit, new heart, and a renewed mind (see 2 Co 5:17 below). This is an unconditional and unilateral action on God's part, through His New Covenant - to be placed in Christ Jesus is entirely of God, not man. That is what makes salvation by grace.
[Col 2:13 KJV]
13 And you, being dead in your sins and the uncircumcision of your flesh, hath he quickened together with him, having forgiven you all trespasses;
[Rom 8:9 KJV]
9 But ye are not in the flesh, but in the Spirit, if so be that the Spirit of God dwell in you. Now if any man have not the Spirit of Christ, he is none of his.
[1Co 1:30 KJV]
30 But of him are ye in Christ Jesus, who of God is made unto us wisdom, and righteousness, and sanctification, and redemption:
[2Co 1:21-22 KJV]
21 Now he which stablisheth us with you in Christ, and hath anointed us, [is] God;
22 Who hath also sealed us, and given the earnest of the Spirit in our hearts.
[2Co 5:17 KJV]
17 Therefore if any man [be] in Christ, [he is] a new creature: old things are passed away; behold, all things are become new.
[2Ti 1:9 KJV]
9 Who hath saved us, and called [us] with an holy calling, not according to our works, but according to his own purpose and grace, which was given us in Christ Jesus before the world began,