The majority of people who take the name Christian believe that God desires to save everybody. Obviously, Arminians believe this for Arminians deny biblical election and reprobation. The Arminian teaches that salvation or non-salvation depends ultimately on the alleged free will of the sinner. This is the case with contemporary Arminians in the world today, and this was the case with the Arminians at the Synod of Dordt (1618-1619). At the Synod of Dordt, the Arminians clearly stated the position called the free offer—that God desires to save everybody. But the Synod of Dordt did not take that position.
The Pelagians and Semi-Pelagians in the early church taught that God desires to save everybody. The Roman Catholic Church also insists that God desires and wishes to save everybody. Arminians and Romanists believe the same thing on these points.
However, many who claim to be Calvinists also believe that God desires to save everybody. Therefore at this point their teaching is the same as the Arminians and the Roman Catholics. They teach that God desires to save the reprobate, although that is not the way they will typically frame it. That lets the cat out of the bag, because it is like saying that God desires to save those whom He has chosen not to save, or God desires to save those whom He does not desire to save because they are reprobate.
Those who do not hold this view that God desires to save the reprobate, commonly called the free offer or the well-meant offer, are told that they cannot truly preach the gospel. If this were true, this is a very serious, even damning, indictment. Then the epithet "hyper-Calvinist" is used. These professed Calvinists, who maintain that there is a desire or wish or will in the very being of God to save the reprobate, teach that this is sincere, for God earnestly wants to save them. They make it clear that this is not just an apparent desire. This is a real desire. In fact, this is an ardent desire. God patiently, longingly wants to save absolutely everybody.
Except, that's wrong and unbiblical.