The EU Parliament has blocked the deal, for now, despite support from EU member states and Commission.
On Wednesday the EU Commission, backed by the member states including Sweden, proposed that the deal should go through. It is seen as an important part of the migration agreement, designed to stop refugees crossing into the EU from Turkey.
But the EU Parliament's political blocs wrote in a joint statement that they would not even hold a vote on the visa-free deal until Turkey has met all 72 conditions that have been laid out. Five of these so-called "benchmarks" are yet to be met. These relate to fighting corruption; cooperating in extradition and working with EUROPOL; protecting personal data; and ensuring "the right to liberty and security, the right to a fair trial and freedom of expression, of assembly and association in practice".
The 67 benchmarks that Turkey has met thus far include many minor items, such as to "Provide information about the conditions and circumstances for the acquisition of Turkish citizenship" and "Use the new Turkish visa stickers with higher security features, stop using stamp visas".
Speaking to Radio Sweden, Swedish MEP Christofer Fjellner, of the conservative Moderate Party, thinks visa-free travel would benefit both Turkey and the EU, and tells Radio Sweden it would be enough if Turkey met most of the criteria.
But Lars Adaktusson, of the Christian Democrats, says Turkey needs to do more work on meeting the proposed standards for human rights before the deal should be allowed to go through.