Some argue that Calvinism and 1 Timothy 4:10 are contradictory because 1 Timothy 4:10 says that Jesus is the “Savior of all people, especially of those who believe.” They argue that this contradicts the doctrine of unconditional election. However, there is a reasonable explanation that does not at all contradict the Calvinist doctrine of unconditional election.
For to this end we toil and strive, because we have our hope set on the living God, who is the Savior of all people, especially of those who believe.1 Timothy 4:10
The most likely interpretation of 1 Timothy 4:10 is that the word “Savior” here does not refer to forgiveness of sins, but rather to God’s common grace for all people. If 1 Timothy 4:10 referred to forgiveness of sins, then that would mean every person in the world would be saved, but Scriptures clearly teaches that this is not true.
The ESV Study Bible says this:
The statement that God is the Savior of all people, especially of those who believe could seem to teach universalism, that every person will eventually go to heaven. However, the rest of Scripture clearly denies this idea (see note on 1 Tim. 2:4). There are several other possible explanations for this phrase: (1) It means that Christ died for all people, but only those who believe in him are saved. (2) It means he is offered to all people, though not all receive him. (3) It means “the Savior of all people, namely, those who believe” (a different translation of Gk. malista, based on extrabiblical examples). (4) It means “the helper of all people,” taking Greek Soter, “Savior,” to refer not to forgiveness of sins but to God’s common grace by which God helps and protects people in need. (5) It means “the Savior of all kinds of people, not Jews only but both Jews and Greeks.ESV Study Bible (Crossway, 2008)
The first option, an Arminian interpretation, should be rejected, since it is unbiblical to say that Jesus died for people who will not be saved. All of the subsequent options are possible, and we would say the fourth option is the most likely interpretation of 1 Timothy 4:10.