Mohammad Javad Larijani - the secretary general of Iran's High Council for Human Rights and a member of one of the country's most influential families - said his office has recommended that the request be granted.
"We are working on that with the security people and judges," he told reporters in Geneva. "We have recommended that the families should be able to see them, and I hope that this will be done."
In a statement, the families welcomed the announcement.
Larijani said the Swiss ambassador in Tehran made the request to his office "about 2-3 weeks ago." Switzerland has represented U.S. consular interests in Iran since Washington and Tehran broke off diplomatic relations following the 1979 Islamic revolution.
Larijani's comment came amid tense relations between Iran and the United States regarding Tehran's nuclear program and its crackdown on public opposition to the nation's disputed presidential election in June.
On Monday, U.S. Secretary of State Hillary Rodham Clinton said the Islamic republic is becoming a military dictatorship, leading Iran's foreign minister to respond that was a good characterization of the United States.
The families of the three detained Americans - Shane Bauer, Sarah Shourd and Josh Fattal - say they were hiking in Iraq's northern Kurdistan region in July when they accidentally crossed the border into Iran.
"This latest report that our visa request is being viewed favorably is encouraging," the families said.
"We have waited over a month already for news about the visas and we have endured over 200 days without communication from Shane, Sarah and Josh," they said. "The lack of contact with them is heartbreaking for us and certainly demoralizing for them."
The families said they hoped the request would be granted soon and they would also be able to meet with Iranian President Mahmoud Ahmadinejad.
Bauer's mother, Cindy Hickey of Pine City, Minnesota, told The Associated Press on Tuesday that the waiting has been "awful" for the hikers' families.
"We don't know if they're getting outside," Bauer said. "There's just so many questions in our heads. ... The nights (for us) are really long with no sleep, little sleep, many nightmares."
Hickey said it's been some 200 days since the families have talked to the hikers. "We want a phone call which we've never gotten," she said. "As a mother I'm very anxious to put my arms around Shane and let him know that we're doing all we can."
Asked about the issue during a news conference Tuesday, Ahmadinejad repeated his previously stated view that the decision to release the Americans was in the hands of the judiciary.
"We are eager to see them released but it depends on the crime they have committed and the judge's verdict," he said, adding that he hoped several Iranians held in U.S. jails would also be freed.
Iran's foreign minister said in late December the three Americans would be tried in court, but he did not say when that would happen or what they would be charged with, other than to say they had "suspicious aims."
Earlier, the country's chief prosecutor said they were accused of spying.
Swiss diplomats last saw the hikers at the end of October, but have not been able to visit them since.
Still, Larijani said the hikers had "full access with the Swiss Embassy and there were several meetings with them."
Larijani said it was "quite possible" the Americans had strayed into Iran by mistake but that "the security people want to be sure this is true" because the area they were in was known for "terrorist activities."
"We should assume that they are innocent," he added.
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Associated Press writers Patrick Walters in Philadelphia and Jeff Baenen in Minneapolis contributed to this report.
