The trip is Mrs Clinton's first official visit to Saudi Arabia
Imposing more sanctions on Iran over its nuclear programme would not be a quick enough solution, Saudi Arabia's foreign minister has said.
Prince Saud al-Faisal said the threat posed by Iran demanded a "more immediate solution" than sanctions.
He spoke in Riyadh alongside US Secretary of State Hillary Clinton, who earlier said Iran was "becoming a military dictatorship".
On Tuesday, Turkey's foreign minister is due in Iran aiming to mediate.
Turkey is a Nato member, and Ahmet Davutoglu is expected to try to promote a deal on Tehran's nuclear programme between Turkey's western allies and Iran's Islamic government.
Kim Ghattas, BBC News, Riyadh
The Saudi foreign minister did not openly back Washington's call for sanctions, but neither did he say Saudi Arabia was opposed to it. He made clear the kingdom wanted a more immediate resolution to the problem rather than a gradual one.
He also said China, a top importer of Saudi oil, did not need to be prodded by Saudi Arabia to know what it ought to do about sanctions against Iran.
Saudi officials are known for using very careful, often opaque, diplomatic language. It sounded as though he was saying the kingdom would not use oil as an incentive to prod China to back UN sanctions against Iran.
But the statement could also be read as a veiled warning - if China failed to back UN sanctions, it risked upsetting its top oil supplier.
Speaking at a joint Riyadh news conference with Mrs Clinton, Prince Saud said: "Sanctions are a long-term solution. They may work, we can't judge.
"But we see the issue in the shorter term maybe because we are closer to the threat... So we need an immediate resolution rather than a gradual resolution."
While the Saudi minister did not detail his vision of a quick solution in public, it is likely that options were discussed behind closed doors in the meeting between Mrs Clinton and King Abdullah, says the BBC's Kim Ghattas, who is travelling with the top US diplomat.
Some regional experts believe neither sanctions nor engagement will work with Iran and have suggested a multi-pronged approach involving intense economic pressure from Iran's neighbours, our correspondent adds.
Earlier, aides to Mrs Clinton - who is on a tour of the Gulf to try to build support for more sanctions on Iran - revealed she would press Saudi Arabia to help persuade China to support a tougher stand against Iran's nuclear ambitions.
China, which can wield a veto on the UN Security Council as a permanent member, is against imposing more sanctions.
Beijing fears a major loss of revenue from investments in Iran, and disruption of oil supplies from a country providing it with 400,000 barrels a day, our correspondent says.
China is a top importer of Saudi oil.
The Saudi foreign minister added that efforts to rid the Middle East of nuclear weapons must also apply to Israel.
Speaking to students at a Qatar university earlier on Monday, Mrs Clinton said Iran's elite army corps, the Revolutionary Guard, had gained so much power they had effectively supplanted the government.
We don't want to be engaging while they are building their bomb
"We see that the government of Iran, the supreme leader, the president, the parliament, is being supplanted and that Iran is moving toward a military dictatorship. That is our view," Mrs Clinton said on her maiden visit to the kingdom.
On Sunday, she urged Iran to reconsider its "dangerous policy decisions".
Mrs Clinton told a conference in Qatar that Tehran was leaving the international community little choice but to impose further sanctions.
The US and its allies fear Iran is attempting to develop a nuclear weapon. Iran insists its nuclear programme is entirely peaceful.
Turkey has already offered to store Iran's nuclear material as part of a swap arrangement agreed last year.
Under terms of that deal, Iran would get medical isotopes from France in return for handing over its own enriched uranium.
Hillary Clinton: "Iran is moving toward a military dictatorship"
Turkey's government hopes its offer to act as a nuclear repository will appeal more to Iran than storing its uranium elsewhere, says the BBC's Jonathan Head in Istanbul.
But Iran is still insisting that any nuclear swap must take place on its own soil.
If no deal can be done with Iran, Turkey will soon be forced to choose between its historically strong alliance with the US and Europe, and its desire for closer friendship with its eastern neighbour, our correspondent adds.
Iran, meanwhile, rejected criticism from the West about its human rights record.
"Iran is becoming one of the predominant democratic states in the region," said Javad Larijani, secretary general of the Iranian High Council for Human Rights.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.