Danny Masterton, 26, a Marine from Muirkirk, Ayrshire is one of the 15 Royal Navy personnel still held in Iran after their capture in the Gulf last week according to local reports.
An MOD statement released on behalf of the family said it was a "distressing" time but thanked relatives, friends and colleagues for their support.
The UK has consistently denied claims that the sailors and marines were in Iranian waters.
Recent reports suggest that Iran has now offered to let UK officials visit the captured Royal Navy personnel, who were seized after boarding a vessel and returning to their boats.
However Iran's foreign minister has said that the stand-off can only be resolved if the UK admits that the RN troops were in Iranian waters. UK defence officials on Wednesday showed satellite data proving they were 1.7 nautical miles inside Iraqi waters when seized.
The Ministry of Defence said it was aware of press reports naming Mr Masterton as the Scottish marine among those captured, but would not confirm his identity.
The family have asked that their privacy is respected.
BBC Scotland reports that Danny was a keen member of the Boys' Brigade from a young age.
Jim Currie, of Muirkirk Boys' Brigade, reportedly told them "The best thing Danny did was to go into the marines. It gave him something to look forward to in life. Unfortunately, he didn't expect this type of thing to happen."
Danny is sadly now caught up in a game being played by the Iranian regime which is seeking to use this incident to humiliate the UK government and armed forces. The fact that the Iranians changed the reported position where they captured the sailors and marines - after it was pointed out that this was inside Iraq waters - shows that this is a crisis that has been manufactured by Iran.
Leading Seaman Faye Turney, 26, from Shropshire, has already been shown apologising for being in Iranian waters and two letters from her have been released. Today there are reports that another British serviceman has apologised for "trespassing" in Iranian waters, in a broadcast on Iranian television. It seems clear that the Iranians intend to milk this situation for all it is worth.
This of course takes us back to the central problem in regard to Iran and the Iranian regime. Is this a country that can be trusted with nuclear development and trusted when it states that it will not develop nuclear weapons? The answer must be a resounding no. This is a country after all who's leader Ahmadinejad is essentially a holocaust denier. We can only hope that the international community takes a lesson from this incident and unites in efforts to thwart Iran's nuclear ambitions.