(CNN) Sitting across from Russian President Vladimir Putin on Tuesday, US Secretary of State Mike Pompeo voiced President Donald Trump's desire for improved relations after earlier warning his counterpart about Russian interference in US elections.

"We'll protect our nation's interests but there are places that our two countries can find where we can be cooperative, we can be productive, we can be accumulative, we can work together to make our two peoples more, and frankly the world, more successful too," Pompeo said in brief remarks prior to his meeting with Putin in the resort town of Sochi.

"President Trump wants to do everything we can and he asked me to travel here to communicate that," Pompeo said.

The Russian President also expressed optimism at warming relations between the two nations.

"As you know, just recently, a few days ago, I had the pleasure of talking with the President of the United States over the phone, I got the impression that the President intends to restore Russian-American relations, contacts, and solve issues that are of mutual interest to us together," Putin said. "We, for our part, have repeatedly said that we would also like to fully restore relations, I hope that now the necessary conditions are being created for this."

"Interference in American elections is unacceptable"

Despite the optimistic remarks, Pompeo had earlier issued a warning about one of the most damning issues between the US and Russia -- Russian election interference. Pompeo noted that the US would not accept meddling in future elections -- a markedly tougher stance than Trump has taken.

"I made clear to Foreign Minister (Sergey) Lavrov, as we've made clear for the past months, that interference in American elections is unacceptable," Pompeo said Tuesday in a joint press avail with the Russian foreign minister.

"If the Russians were engaged in that in 2020, it would put our relationship in an even worse place than it has been, and I encouraged them not to do that, that we would not tolerate that," Pompeo continued. "We've said this not only about the Russians but about other countries as well. Our elections are important and sacred and they must be kept free and fair and with no outside country interfering in those elections."

Pompeo is the first high-level US official to Russia since the release of the Mueller report and his message to his Russian counterpart comes in contrast to Trump's soft-handed approach. During that recent phone conversation with Putin, Trump did not warn his Russian counterpart not to meddle in US elections. The two leaders also discussed the Mueller report, which the US President described again as a "hoax." Following the phone call, Trump took to Twitter to emphasize his optimism about the future of the US-Russia relations.

"Tremendous potential for a good/great relationship with Russia, despite what you read and see in the Fake News Media," Trump wrote. "Look how they have misled you on 'Russia Collusion.' The World can be a better and safer place. Nice!"

The Russian foreign minister took a similar stance on the report, which documented extensive efforts by Russia to influence the 2016 election.

"A report was published by special counsel Robert Mueller and we hope that this tumultuous situation will die down and we can finally move on to building more professional, constructive dialogue between Russia and the US," Lavrov said.

And Putin seemed to praise the work of special counsel Mueller, saying, "Despite the exotic nature of the work of special counsel Mueller, we must give him credit."

"In general, he conducted an objective investigation and confirmed the absence of any trace and any kind of collusion between Russia and the current administration, which we characterized as nonsense from the get go," he said.

"We fundamentally do not seek a war with Iran"

On the issue of Iran, Pompeo said he "made clear that the United States will continue to apply pressure to the regime in Tehran until its leadership is prepared to return to the ranks of responsible nations that do not threaten their neighbors and do not spread instability or terror."

However, Pompeo would not comment on a report that the US may be readying to send hundreds of thousands of troops to the region amid escalating tensions with Iran. He said it was a matter for the Department of Defense.

"We fundamentally do not seek a war with Iran," Pompeo said, but noted that "if American interests are attacked, we will certainly respond in an appropriate fashion."

Lavrov said that, when it comes to the tensions over the Iran nuclear deal, "I hope very much that common sense will triumph and that rumors of a US troop build-up in the region... are unfounded because the region is overburdened with conflict as it is."

Pompeo said he did not have any information that he could share about incidents involving four oil tankers near United Arab Emirates territorial waters on Sunday.

"We are working diligently to get answers to what caused those ships to have the problems that they have today," Pompeo said.

The UAE has called the incidents, which affected two Saudi, one UAE, and one Norwegian tanker, as a "sabotage attack." The UAE has requested US technical assistance in investigating the incident, according to a US official. Iran's Foreign Ministry spokesperson Seyyed Abbas Mousavi said on Monday that the incidents were "alarming and regrettable," and requested further information on the alleged sabotage.

On Monday, Pompeo canceled a stop in Moscow and instead went to Brussels to brief European allies on intel about Iran.

US "made clear" to Russia its view on Venezuela

Pompeo said he made the US position on Venezuela clear to Lavrov -- that the US wants "every country that's interfering in Venezuela to cease."

"So whether its Iranian forces, or Chinese or Cubans, the Trump administration position is that they all need to cease having an impact in supporting embattled Venezuelan President Nicolas Maduro and allowing the Venezuelan people not only to get their democracy back but give them an opportunity to rebuild this country that has tremendous wealth," Pompeo said.

Pompeo recently accused Russia of preventing Maduro's departure from the country. The claim was directly contradicted by Trump, who said following his conversation with Putin that the Russian President had told him that Russia "is not looking at all to get involved in Venezuela other than he'd like to see something positive happen for Venezuela."

Lavrov said he told Pompeo that the Venezuelan people should decide their future and that Russia has acted "without any kind of ultimatums or conditions and our position essentially is that democratic forces cannot be stopped."

Arms control "very much on President Trump's mind"

Pompeo stressed the issue of arms control, saying it was "very much on President Trump's mind."

"(The) President wants serious arms control that delivers real security to the American people," Pompeo said. "To achieve these goals, we'll have to work together and it would be important, that if it's possible, we get China involved as well."

The two diplomats also discussed a series of other bilateral issues during their hour-and-a-half-long meeting, including maintaining pressure on North Korea and the escalation of violence in Idlib, Syria.

Pompeo said he raised the issue of Russian annexation efforts in Crimea and noted that the US would "welcome the release of the Ukrainian crewmen detained near the Kerch Strait last year."

However, asked later whether it was still a precondition that Russia must release the Ukrainian sailors before Trump would meet with Putin -- as they are expected to do at the G20 Summit in June -- Pompeo did not answer.