Ms Marin said both nations were "fully prepared" to become Nato members.
Turkey's support is proving especially difficult to win as it demands extraditions of terror suspects and a tougher line on Kurdish militants.
Ms Marin said Finland had met these conditions and that Turkish President Recep Tayyip Erdogan had indicated he was more satisfied with Finland than Sweden.
But she said the two countries should join simultaneously. They applied last May after the Russian invasion of Ukraine.
"Hopefully the ratification process will go as fast as possible," Ms Marin said at the World Economic Forum in Davos.
"I think the process should have been faster. We both tick all the boxes when it comes to Nato, we are fully prepared to become Nato members and there shouldn't be any obstacles in the way.
"Hungary have said that they will ratify as soon as their parliamentary period starts, hopefully very soon. Turkey, that’s another matter."
The two applicants signed an agreement last year promising to meet Turkish demands on extraditions, arms exports and counter-terrorism.
But Turkey has said it wants to see results before it lifts its veto. Sweden last week said Turkey had introduced new demands which it could not meet.
Ukraine also aspires to Nato membership but senior alliance figures see that as a question for another day while the war with Russia rages on.
"If a country is in the middle of war it cannot become a member of Nato," Ms Marin said.
However, she said if Ukraine had been admitted to Nato earlier, "there wouldn't be a war now in Ukraine".