"I am considering meeting with my counterpart but there has been no decision," Mattis told journalists at the Pentagon.

"I'm all for re-opening communications," he added. "It's most important that we talk with those countries we have the largest disagreements with."

In Johannesburg, Russian President Vladimir Putin said he was "ready" to travel to Washington to see Donald Trump, and revealed he had invited the US leader to Moscow, shortly after their July 16 summit in Helsinki.

The White House said Friday that Trump was "open" to visiting the Russian capital, though it said earlier this week that any new meeting between the two leaders would likely not happen until 2019.

Trump's performance at the summit earned howls of criticism at home, where he was seen as being too conciliatory towards Putin.

Mattis said there were "no policy changes" with respect to Russia in the wake of the summit.

On Tuesday, Mattis said there would be no cooperation with the Russian military in Syria -- at least for now.

Russia and the US are conducting separate military campaigns in the war-wracked country. They operate a special hotline to make sure there are no mishaps involving the two sides' ground forces or planes.

"We will not be doing anything additional until the Secretary of State (Mike Pompeo) and the president have further figured out at what point we are going to start working, alongside our allies, with Russia in the future," Mattis said at a press conference in California.

According to the Pentagon, the last ministerial-level contact between US and Russian defense ministers dates back to a telephone call between Shoigu and Mattis's predecessor Ash Carter in September 2015.

The last face-to-face meeting dates back to October 2013 between Shoigu and Chuck Hagel, shortly before Russia annexed Crimea.