Hamza Yalcin was arrested on August 3 at Barcelona's El Prat airport and is being held while a court decides whether to extradite him or not, Spanish police told AFP.
According to media rights group Reporters Without Borders (RSF), Turkey accuses him of "terrorism."
"The foreign ministry has asked to be allowed to visit the Swedish national detained in Barcelona, Spain. The foreign ministry has made sure that he has a lawyer and is working to clarify the charges," said Swedish Foreign Minister Margot Wallstrom.
"Even if we cannot get involved in the legal process of another country, we can ensure that a process is conducted in accordance with general principles of law. Of course that becomes more difficult when it comes to countries who do not live up to the rule of law and do not share our basic values."
According to RSF, the arrest warrant is an attempt "to try to silence criticism of the Turkish regime," which activists accuse of detaining dozens of journalists under the state of emergency imposed after last year's coup attempt.
The group says that since 1984, Yalcin has lived in exile in Sweden, where he has written for Odak Dergisi, an online magazine critical of Ankara.
Jonathan Lundqvist, head of RSF Sweden, criticised the arrest as an attempt by Turkish President Recep Tayyip Erdogan "to extend his power beyond the country's borders."
"He wants to show that he can get at critics even if they are not in the country. This is an abuse of international police cooperation, which risks having major consequences," he said in a statement.
The arrest comes as alarm grows over press freedom in Turkey under Erdogan, with foreign reporters also being caught up in the crackdown.
Turkey ranks 155 on the latest RSF world press freedom index, below Belarus and the Democratic Republic of Congo, after dropping four places from its 2016 ranking.
According to the P24 press freedom website, there are 162 journalists behind bars in Turkey, most of whom were detained after the July 2016 coup bid.