Erdogan was to hold three-way phone talks on the crisis later Tuesday with French President Emmanuel Macron and Qatar's emir, Sheikh Tamim bin Hamad Al-Thani.
Smith's departure comes with Washington sending mixed signals over the Gulf crisis, which saw Saudi Arabia and several of its allies cut ties with Qatar claiming that Doha supported extremist groups.
The Gulf states on June 5 ordered Qataris to leave within 14 days and also banned their own citizens from travelling to the emirate.
Late last month, Putin met the Saudi Deputy Crown Prince Mohammed bin Salman in Moscow for talks on the Syrian conflict as well as cutting oil production.
It says Sharif's visit reaffirmed Pakistan's strong commitment for Saudi Arabia's territorial integrity and sovereignty, and quotes King Salman as saying the joint fight against terrorism is in the interest of all Muslims countries.
In a statement Tuesday, the Turkish military said the country sent a three-person team Monday to scout and coordinate preparations for deployment.
Qatar's border with Saudi Arabia is its only land frontier, and the closure of Saudi, Bahraini and Emirati airspace to Qatar Airways jets has disrupted its normal routes.
He said Pakistan has good relations with Qatar, Saudi Arabia, UAE and other Gulf countries and it should play a positive role to end the current standoff developing in the region.
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Oman is a member of the Gulf Cooperation Council but takes a relatively independent diplomatic approach from the other five states in the bloc, especially on relations with Iran, a sticking point in the dispute with Qatar.
Russian President Vladimir Putin and Saudi King Salman discussed the crisis on Tuesday, with Putin warning that isolating Qatar would make finding a peaceful end to the war in Syria more hard.
Pakistan's prime minister and army chief are heading to Saudi Arabia, where they hope to help ease tensions related to the Gulf diplomatic crisis. These governments accused the tiny country of destabilizing the region by supporting terrorist groups - an allegation Qatari officials deny.
Last week, Saudi Arabia, Egypt, the United Arab Emirates and Bahrain severed ties with Qatar.
On Tuesday, Republican Senator Rand Paul introduced a resolution - alongside two Democrat senators - to block a major United States arms sale to Saudi Arabia.
The isolation of Qatar - labelled a "blockade" by Doha political leaders - also has potentially severe economic implications.
Qatar Airways on Monday called on the UN's aviation body to declare a Gulf boycott against the carrier "illegal" and a violation of a 1944 convention on global air transport.
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