He named the three as Shane Bower, Sara Short and Joshua Steel, while Shaun Gabriel Maxwell stayed behind in their hotel in the autonomous Kurdish region's second-largest city of Sulaimaniyah.
Falah Mustafa, foreign policy chief for the Iraqi Kurdistan government, said: "They were interested in going up the mountain and after that they walked down the other side, which is the Iranian side.
"Because they don't know the area, they entered Iran's lands."
There is no clear border marker between Iran and Iraq at Ahmed Awa.
A spokeswoman for the US embassy in Baghdad said: "We've seen the reports and are looking into it but can't confirm anything at this time."
An official at the Pentagon in Washington said no US military personnel were involved.
The arrests came after the three visited the mountainous resort region of Ahmed Awa, about 90km northeast of Sulaimaniyah.
Ahmed said: "The [Kurdish] tourist police in the area asked them not to climb the mountains because the Iranian border was very close.
"On Friday, they went close to the mountains, and climbed them. Then they called their friend in the hotel telling him that they were arrested by Iranian forces at the border."
"Shaun was in the hotel and he called the US embassy in Iraq to tell them about this information, and the Americans came to the hotel and took him."
Ahmed said the group had originally been in Syria before going to Turkey and eventually crossing the Turkey-Kurdistan border.