Samuel Ofosu-Ampofo, national chairman of the National Democratic Congress (NDC), is a former member of parliament and government minister.
His arrest comes one week after the abduction of two Canadian women, who remain missing.
"The Criminal Investigation Department is investigating various acts of kidnapping and causing unlawful damage, in which credible and actionable intelligence makes Mr. Samuel Ofosu-Ampofo, the National Chairman of the National Democratic Congress (NDC), a suspect," a police statement said on Tuesday.
Ofuso-Ampofo was arrested on Tuesday afternoon in Accra after he refused to cooperate with police investigations, the statement said.
"Mr. Samuel Ofosu-Ampofo is currently in lawful police custody in respect of the ongoing investigations," police said, calling on the public to remain calm.
The Canadians, charity volunteers aged 19 and 20, were abducted June 4, in Kumasi, Ghana's second-largest city, some 200 kilometres (125 miles) northwest of the capital Accra.
The police, who have not released the names of the women, are working with Canadian authorities to find them.
Abductions and violent crime towards foreigners are rare in Ghana.
But earlier this year, local media quoted President Nana Akufo-Addo as warning that action must be taken to make sure kidnapping "doesn't become a feature of our society."
Last year, three Ghanaian women were kidnapped in the southern port of Takoradi.
In April, an Indian man was reportedly abducted, also in Kumasi, by an armed gang demanding a cash ransom. He was swiftly rescued by police.
The Estonian consul was also briefly abducted in April, but released soon after.