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A New Zealand couple who went silent after entering Iran almost four months ago say they are "extremely relieved" to have finally returned home.

Topher Richwhite and Bridget Thackwray arrived there in July as part of a tour they were documenting for their social media page, Expedition Earth.

But fears were raised after they abruptly stopped posting on their Instagram account.

It is unclear what happened after they entered Iran.

In a video since deleted from their page, Mr Richwhite - the son of one of New Zealand's richest men - said the couple had been stopped at the Iranian border, where the vehicle was inspected.

Their page - usually filled with colourful updates of their adventures - fell silent soon after a terse meeting with border officials.

New Zealand media reported that the newlyweds had not been imprisoned, but that they were unable to leave Iran and that their movements were being restricted and monitored by regime officials.

Tehran insisted that the pair had not been imprisoned or detained, and the government in Wellington was reluctant to comment while discussions were ongoing - in fear of jeopardising their release.

In September, New Zealand Foreign Minister Nanaia Mahuta raised the issue with her Iranian counterpart, and the couple were finally released on Tuesday after months of negotiations.

"We assisted two individuals to secure the necessary documentation in order to leave," Ms Mahuta said on Wednesday. "During the period of time we were working alongside them, their movements were restricted. They were monitored but they were safe and well."

In a short statement released on Wednesday, the couple said they were "most grateful to all those who have supported us over recent months and thank them sincerely for all their help during this challenging chapter".

"We are safe and well, however we would ask for privacy for ourselves and our families over the coming days while we collect our bearings and enjoy being reunited with our families," the statement added.

Despite some speculation in local media, New Zealand's government has insisted that it had not given Tehran anything to secure their release.

"There is no deal, there's nothing in return. Absolutely not ... I can absolutely guarantee you no deal has been done," deputy Prime Minister Grant Robertson told reporters.

Westerners have frequently been detained by the Iranian regime, which seeks to use them as bargaining chips to secure concessions from governments which have sanctioned it.

As a result, most Western countries - including New Zealand - advise against travelling to Iran.

Some in New Zealand's media have questioned why the couple ignored that advice. Mike Yardley, a columnist with the Newshub media outlet, questioned why "so-called travel influencers were oblivious to that status - or spurned the official advice and threw caution to the wind".

"Either way, self-responsibility was sadly lacking," he added.

On Wednesday, the New Zealand government updated its travel advice for Iran, advising all its citizens to leave the country.

In March, the UK agreed to pay a historic debt of £400m ($464m) to secure the release of Anoosheh Ashoori and Nazanin Zaghari-Ratcliffe.

Ms Zaghari-Ratcliffe was first detained by Tehran on trumped up espionage charges in 2016.