The six French passport holders, five men of Turkish descent and one of Algerian descent, were detained on Saturday evening after arriving at Larnaca airport on a low-cost flight from Basel, Switzerland, police said.
Police said the suspects were aged between 25 and 33.
Justice Minister Ionas Nicolaou told state radio that at least three of the six appeared to have "strong links to terrorist groups".
"There were suspicions against these specific individuals upon their arrival at the island," Nicolaou said.
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"The other three also appear to have some sort of link. They will probably be deported back to Switzerland at some point today."
Once they return to Switzerland the suspects will be handed over to French authorities, Swiss federal police told AFP.
"This has been discussed with French authorities, they are the ones in charge of the case and they will take care of the suspects once they arrive," a spokesman said.