The free offer of the gospel is the sincere, universal proclamation of God's command for all men to repent and believe in Christ for salvation, accompanied by the promise that all who believe shall have eternal life. While Reformed traditions agree on the sincerity of this call, they emphasize that it is only through the effectual work of the Holy Spirit that the elect are made willing and able to embrace the grace offered.
The gospel is a gracious announcement that the finished work of Christ is sufficient for the salvation of all who believe. According to the Canons of Dort — Head 2, Article 4, the promise of life is to be published to all nations and persons without distinction. This call is not a hypothetical offer, but a genuine command from the Sovereign God, which requires of sinners faith in Christ (1689 LBCF Ch.7 §2). We must remember that you are more sinful than you ever dared believe, yet because of the offer of the gospel, you are more accepted and loved in Christ than you ever dared hope.
While the gospel is offered universally, it is the Holy Spirit alone who makes the call 'effectual.' The Westminster Larger Catechism — Question 67 teaches that God, out of His special love for the elect, works by His almighty power to enlighten the mind and renew the will. Without this internal work of the Spirit, our hearts remain dead in sin and incapable of responding to the gospel. Thus, our obedience to the call of the gospel is not the cause of our acceptance, but the fruit of God's sovereign grace, which alone makes us willing and able to come to Christ.