Turner, 73, has had a house in Zurich since the mid-1990s, but makes frequent visits to the United States.
Swiss officials have already approved her request for citizenship, according to gossip site TMZ, citing a Swiss newspaper.
Zurich is a well known Swiss tax haven that has a large American expatriate community, mostly made up of tax cheats or those trying to avoid U.S. income taxes. One of the most famous was fugitive financier Marc Rich.
The international commodities trader and hedge fund manager was indicted in 1983 for failing to pay tens of millions of dollars in U.S. taxes and for trading with Iran during the 1979 hostage crisis when the United States had cut off all relations with the radical Islamic regime.
He was in Switzerland at the time and chose to stay there. The Swiss do not extradite U.S. citizens for tax crimes. On Jan. 20 2001, hours before leaving office, President Clinton granted Rich a full presidential pardon.
As part of the deal, Rich agreed to pay his outstanding taxes, but the payment amounted to pennies on the dollar. The pardon caused an uproar because Rich’s ex-wife, New York socialite Denise Rich, 68, had contributed millions of dollars to the Democratic Party and Clinton Library.
Rich, born to Jewish parents in Belgium, came to the United States in 1941 as a toddler, after the country graciously accepted his parents, who were fleeing Nazi persecution.
In his request for a pardon, he claimed to have been an informant for Mossad, the Israeli Intelligence Agency.
Wife Denise Rich, worked as a songwriter mostly in the 1980s, and counted Turner among her close friends. She has written songs for Phyllis Hyman, Natalie Cole, Celine Dion, Jessica Simpson, Marc Anthony, Patti LaBelle, Chaka Khan and Diana Ross.
Perhaps not surprisingly, Rich renounced her U.S. citizenship in November 2011 to become an Austrian citizen. She acknowledged she made the move at least partly to avoid paying U.S. taxes.