The phrase "close with Christ" is extensively used by the puritans. God calls the sinner, and the sinner is to close with Christ (i.e. do that which God desires them to do in exercising faith). Isaac Amborse says, "The gospel and God’s Spirit clearly revealed the fullness of God’s mercy in Christ, but then the whole soul (both the mind that discovers mercy, and hope that expects it, and desire that pursues it, and love that entertains it, and the will that rests on it), gives answer to the call of God in it...The mind says, Let me know this mercy above all, and desire to know nothing but Christ and Him crucified. Let me expect this mercy (says hope), that belongs to me, and will befall me. Desire faith, Let me long after it. O, says love, let me embrace and welcome it. O, says the heart, let me lay hold on the handle of salvation, where we will live, and where we will die at the footstool of God’s mercy. Thus all go, mind, hope, desire, love, joy, the will, and all lay hold upon the promise, and say, Let us make the promise our prey, let us prey upon mercy, as the wild beasts do upon their provision. Thus the faculties of the soul hunt and pursue this mercy, and lay hold thereupon, and satisfy themselves therein. (Isaac Ambrose, Regeneration and the New Birth).
Just about every puritan work on salvation and conversion speaks about closing with Christ - coming over the finish line of that which is necessary to be saved.
Thomas White said, "In the second place, this translation is wrought by a powerful inclination and conversion of the will to close with Christ upon his own terms."
Thomas Doolittle said, the elect "close with Christ by faith unfeigned, and be partakers of eternal life through him."
Francis Roberts said, "Christ applies this obtained redemption to us immediately by his Spirit, effectually working faith and all saving grace in us, so that we may close with Christ, have union to him, and communion with him, (Titus 3:4-7; Eph. 2:8-9; Gal. 5:22; Ezek. 36:26-27; Deut. 30:6; Eph. 1:13-14; 1 John 5:12; 1 Cor. 1:30). (The Natural Man Directed to Jesus Christ).
The Christian's Great Interest by William Guthrie (readily available online) has a wonderfully clear and succinct explanation of closing with Christ. It is found in chapter 2, The Duty of Closing with God's plan of saving sinners by Christ Jesus.
"So then, to close with God's device of saving sinners by Christ Jesus, is to quit and renounce all thoughts of help or salvation by our own righteousness, and to agree unto this way which God has found out: it is to value and highly esteem Christ Jesus as the treasure sufficient to enrich poor sinners; and with the heart to believe this record, that there is life enough in Him for men: it is to approve this plan and acquiesce in it, as the only way to true happiness: it is to point towards this mediator, as God holdeth Him out in the gospel, with a desire to lay the stress of our whole state on Him. This is that which is called faith or believing, the 'receiving of Christ,' or 'believing on His name.' (John 1:12.) This is that 'believing on the Lord Jesus Christ,' commanded to the jailer for his safety. (Acts 16:31.) This agreeth to all the descriptions of justifying faith in the Scripture. This answers to the type of looking to the 'brazen serpent lifted up in the wilderness' (John 3:14, 15); and this is supposed in all those ordinary actings of faith to which promises are annexed in the Scripture; and will be found in all who have got the new heart from God, and it will be found in none else."