Lavrov said Moscow was ready to act "with the consent and the interest of the parties involved" to help resolve the diplomatic row.
"They have to end that funding and its extremist ideology in terms of funding", he said, lumping Qatar in with other nations which he said must "stop teaching people to kill other people".
Qatar Petroleum's statement comes during the biggest diplomatic crisis to hit the Gulf states in years.
(This refiled version of the story adds dropped title in paragraph two).
"Qatar is accused of having a hidden relationship with Iran, but its relations with Iran are clear, transparent and time-tested", said Sheikh Mohammed bin Abdulrahman al-Thani, noting that the UAE does more trade with Iran than Qatar does.
Tillerson on Friday urged Qatar to be "responsive" to the concerns of its neighbours as he called on Doha to halt financial and other support for terrorists. "With over eight million Indian expatriates living and working in these countries, we have vital stakes in the regional peace and stability", the ministry said in a release.
On Friday, Saudi Arabia and the United Arab Emirates escalated the dispute by issuing a blacklist of 59 individuals, including prominent Qatari politicians and businessman, and 12 organizations whom they claim are aiding terrorism.
Trump Says He Is '100%' Willing To Testify Under Oath About Comey
Kasowitz also maintained that the testimony made clear that Trump "never, in form or substance, directed or suggested that Mr. Senator Susan Collins said it was possible Comey misremembered or misinterpreted some of their exchanges.
Tillerson praised efforts by Kuwait to mediate the crisis and said Washington would support the emir of Kuwait.
Qatar acknowledges its links to these groups but denies that it supports terrorism.
Despite Mr Tillerson's plea for "no further escalation", Mr Trump's sharp comments are likely to further embolden Saudi Arabia and others in their bid to isolate Qatar. The list included the International Union of Muslim Scholars' Egyptian chairman, Yousef al-Qaradawi, and Abdullah bin Khalid, a former interior minister of Qatar.
US Secretary of State Rex Tillerson and Russian Foreign Minister Sergey Lavrov have called for dialogue over the Qatar-GCC dispute, as a blockade imposed by three Arab states on the country entered its seventh day.
The moves taken against Doha include banning Qatar Airways from airspace and closing Qatar s only land border with Saudi Arabia.
The Arab states have also ordered Qataris out within 14 days.
Turkey will send warplanes and warships to Qatar after an initial deployment of troops, the mass-circulation Hurriyet newspaper said on its website.
- Rafael Nadal remporte son dixième Roland-Garros
- Election results 2017: The Jeremy Corbyn factor
- Ariana Grande's Manchester benefit show sells out in minutes
- UK Politicians Hastily Thinking of Coalition Options After Snap Election
- Theresa May's DUP deal: Hundreds of thousands back call to block agreement
- Red Sox announcer Remy says Tanaka shouldn't get translator
- Partial results: Most voters in Puerto Rico back statehood
- Sterling shaken by United Kingdom election shock
- This man's moustache won the BBC Question Time debate
- Penguins a Game 6 win from defending Cup title