Reza Seifollahi, an official in the country's Supreme National Security Council, was quoted by Iranian media as saying that the perpetrators of the attacks were Iranian nationals.
Several attackers raided the shrine of the late founder of the Islamic Republic, Imam Khomeini, South of the capital Tehran.
Intelligence Minister Mahmoud Alavi told IRNA on Thursday that it is clear Saudi Arabia supports terrorist groups but it is too soon to say whether it was involved in the Parliament and mausoleum strikes.
Police officers stand guard as vehicles drive past Iran's parliament building in Tehran, Iran, Thursday, May 8, 2017.
Middle East expert Catherine Shakdam told RT that she is confident there is "more than a connection" between the attacks and Saudi Arabia.
Trump has repeatedly accused Iran of sponsoring "terrorism".
Iranian authorities warned past year that the country faces a significant threat from ISIS, which promotes an ultra-conservative strand of Sunni Islam that analysts have described as "medieval".
The attacks were the first claimed by ISIS, the hardline Sunni Muslim militant group, and released a video purporting to show gunmen inside the parliament building and one body, apparently dead, on the floor.
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According to the director of the Crisis Research Institute, Mark Almond, it is highly possible IS was, as it claims, behind the attacks in Iran as the group is interested in stirring up a Sunni-Shiite conflict.
Egypt condemned "in the strongest terms" Wednesday's twin terrorist attacks in Iran that killed at least 12 people and injured more than 40, the foreign ministry said. One terrorist committed suicide by blowing himself up.
But by mounting a successful attack, IS could claim a major coup against a traditional foe that other Sunni jihadist groups, including its rival al-Qaeda, have failed to target in the past.
The intelligence ministry said security forces had arrested another "terrorist team" planning a third attack.
But Rouhani said Iran would be more united and more determined in the fight against regional terrorism and violence.
A separate terrorist group in the country has been detained before it managed to carry out any terror attack.
"The terrorists had explosives strapped to them and suddenly started to shoot around", said the shrine's overseer, Mohammadali Ansari. IS fighters have looted and destroyed historical artefacts in Syria and Iraq in an attempt at "cultural cleansing", and thousands of women from religious minorites have been enslaved.
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