The two-day visit comes amid a thaw in ties between Tehran and six-nation Gulf Cooperation Council (GCC) since the election of Iran's moderate President Hassan Rouhani in June 2013.
Sheikh Sabah, on his first visit to Tehran as head of state, flew in at the head of a high-level delegation including the foreign, oil, finance, commerce and industry ministers.
Tehran says the visit would "open a new page" in ties between the neighbours across the Gulf.
The visit will also focus on controversial regional issues, including Iran's military involvement in Syria, the situation in Iraq and Egypt, and the Middle East peace process, Kuwaiti officials said.
Relations between Iran and the Gulf, namely signs of rapprochement between regional power brokers Saudi Arabia and Iran, will also be discussed during the visit, Kuwaiti foreign ministry under-secretary Khaled al-Jarallah told Al-Hayat newspaper.
He said Kuwait, which currently holds the rotating presidency of the GCC -- which also includes Bahrain, Oman, Qatar, Saudi Arabia and the United Arab Emirates (UAE) -- has balanced links with Tehran and is willing to mediate between Riyadh and Tehran.
Riyadh is also at odds with Iran over the Syria war, in which Tehran backs the government and Saudi Arabia the rebels, as well as its involvement in Iraq, Bahrain and other countries in the region.
Last month, Saudi Foreign Minister Prince Saud al-Faisal invited his Iranian counterpart Mohammad Javad Zarif to visit Riyadh. He accepted but no date has yet been set.
Saudi Arabia has also invited Iran to attend a two-day meeting of the Organisation of Islamic Cooperation that opens on June 18 in Jeddah, with Tehran welcoming the invite as a "friendly" gesture.
In December, Zarif toured Kuwait, the UAE, Oman and Qatar, but skipped Bahrain and Saudi Arabia.
Kuwait's ambassador to Tehran, Majdi al-Dhafiri, told Kuwait's official news agency KUNA that Sheikh Sabah and Rouhani will discuss a number of "strategic projects" useful for the whole region. He did not elaborate.
Get Breaking news, live coverage, and Latest News from India and around the world on NDTV.com. Catch all the Live TV action on NDTV 24x7 and NDTV India. Like us on Facebook or follow us on Twitter and Instagram for latest news and live news updates.