A marked police van was parked outside the former Scottish leader's home in Glasgow, which was sealed off with blue and white police tape.
"The man is in custody and is being questioned by Police Scotland detectives," the force said.
The police investigation is looking at what happened to more than $750,000 raised by Scottish independence campaigners in 2017 that may have been used for other purposes.
The party's accounts, published by the Electoral Commission, showed that at the end of 2018 the party had about $512,000.
The SNP has not commented as the investigation was still ongoing but said it had been cooperating with the probe.
Sturgeon was recently replaced as SNP leader and Scotland's First Minister by Humza Yousaf in a contest that exposed deep divisions within the party.
Yousaf called the arrest a "difficult day for the party" and said that he wanted to reassure SNP members over party transparency and finances.
"The news this morning, it's challenging and it's difficult," he said.
Murrell, who had run the SNP for more than two decades and has been married to Sturgeon since 2010, resigned last month after accepting the blame for misleading the public about a plunge in the number of party members.
Sturgeon also stood down as the leader of Scotland's semi-autonomous government last month after eight years in power, saying she had become too divisive to lead the nation to independence.