Manouchehr Mottaki, Iran's foreign minister, said on Wednesday that Turney would be released "within a day or two".
He has since said Britain must admit that its 15 soldiers entered Iranian waters to help end a standoff over their capture.
Mottaki's statement comes amid escalating tensions, highlighted by Iranian television footage of the Britons showing the only woman captive saying her group had "trespassed" into Iranian waters.
Britain denounced the video as unacceptable and froze most dealings with Tehran.
Mottaki said that if the alleged entry into Iranian waters was a mistake "this can be solved".
"But they have to show that it was a mistake - that will help us to end this issue," he said.
"Admitting the mistake will facilitate a solution to the problem,'' he said late on Wednesday.
It was the first time that Iran has publicly suggested a way to resolve the crisis.
Britain maintains that its troops were in Iraqi waters when they were ambushed, and released documents on Wednesday to back up the claim.
Mottaki also said that Tehran had agreed to allow British officials to meet the captured sailors.
"The British may prefer diplomacy... but they don't like being humiliated"
Surfdog1958, Baton Rouge, US
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"We have accepted that [British request], there is no problem. Measures are under way. They can meet them," he said.
That concession did not appear to assuage British outrage.
Tony Blair, the prime minister, told parliament on Wednesday that "there was no justification whatever ... for their detention, it was completely unacceptable, wrong and illegal".
"We had hoped to see their immediate release; this has not happened. It is now time to ratchet up the diplomatic and international pressure in order to make sure the Iranian government understands its total isolation on this issue," he said.
|"Admitting the mistake will facilitate a solution to the problem'' |
Manouchehr Mottaki, Iran foreign minister
Mottaki also said earlier reports that he had said Turney could be freed on Wednesday or Thursday were incorrect, saying: "I was probably misquoted."
But whether by Thursday or "soon", the British foreign ministry said that was "not good enough", repeating a call for the release of all 15 crew members.
"Obviously the release of one person, one of the military personnel and not the others, is not good enough," a ministry spokesman said.
Britain will seek a UN Security Council endorsement of a statement on Thursday that would "deplore" Iran's detention of 15 of its sailors, according to a draft text.