Ali Larijani, of the Supreme National Security Council, told state radio the issue would be resolved if the two countries "continue this path".
The UK Foreign Office said talks were continuing with a "more positive feel".
Iran accuses the crew of straying into its waters on 23 March. The British insist they were in Iraqi territory.
Mr Larijani, who is Iran's chief nuclear negotiator and head of the powerful security council, hinted that fresh talks between his country and Britain had opened on Friday.
He told Iranian state radio: "The British government has started some diplomatic talks with the Iranian Foreign Ministry. They are only at the beginning.
"If they continue this path they can change the situation logically... in such a way that the issue is resolved."
But UK officials later said that no new negotiations had begun - although talks had been continuing since the sailors and marines were captured.
Mr Larijani has previously said he was committed to solving the crisis through diplomacy, and that putting the crew on trial was "not a priority".
But he insisted in an interview with Channel 4 News on Monday he was "100%" sure that the navy crew had been in Iranian waters when they were picked up in the northern Gulf on 23 March.
"A guarantee must be given that such violations will not be repeated," he said.
The UK insists they were in Iraqi waters after returning from searching a merchant vessel.
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