"We are concerned about the Turkish incident in Northern Syria", Tillerson said while on a trip to London, according to a pool report supplied to Reuters.
Mr Tillerson highlighted Iran's ballistic missile programme not being covered by the nuclear deal as one of the specific concerns.
It was a notable softening of language on the issue compared with that used by the USA president, who has called the deal "an embarrassment to the United States".
"We think we can do that".
"We recognize and fully appreciate Turkey's legitimate right to protect its own citizens from terrorist elements", he said.
Later, Tillerson went further and told reporters that a team of United States officials was travelling to Europe to "meet with counterparts, so we'll see what we can get done".
The State Department said it had not been decided whether he would formally open the building.
US Vice President Mike Pence also weighed in on the nuclear deal Monday. Iran has said it will no longer be bound by the terms of the deal if that happens. Trump earlier this month criticized the move to new diplomatic premises as part of a bad deal agreed by the administration of Barack Obama.
"Wanted me to cut ribbon - NO!" The decision had been agreed in 2008 during the presidency of George W. Bush.
The main objective of Rex Tillerson's visit to Warsaw is to strengthen the strategic partnership with Poland, a high-ranking representative of the US Department of State has said.
France And UK Work Through Border Issues In Summit
Instead she stressed the need to clamp down on people smugglers and take other measures to stop migrants from getting to Calais. Amid a sudden hailstorm, Mr Macron and Mrs May inspected the honour guard before taking a salute from soldiers on horseback.
Earlier this month, Trump said he had canceled a trip to perform a ribbon-cutting ceremony at the embassy due to an Obama-era "bad deal", in which the old embassy had been sold off for "peanuts". The new embassy was constructed with the proceeds from sales of USA government properties and leases in London. Tillerson was nevertheless introduced to a number of diplomats as he was given a tour of the building.
"The embassy actually, is gonna really work", Johnson said. It is absolutely right that we should welcome him here, and it is time for Labour to end their Spartist agitations.
At a press conference, Mr Tillerson said he treasured his relationship with Mr Johnson.
Trump "is the elected president of the world's most powerful democracy" and resisting his visit to the United Kingdom risks "damaging the national interest", Johnson added. The invitation was accepted, but no date for the visit has been set.
Mr Tillerson also had a meeting with National Security Adviser Mark Sedwill, with Iran and North Korea on the agenda.
Trump's cancellation of his trip has raised questions in Britain over the links between Washington and its closest traditional ally in Europe - widely called the "special relationship".
"We spend a lot of time talking about the world's problems", he said.
But, he stressed, "it is important that we do that in parallel and don't vitiate the fundamentals of the Iran nuclear deal". We treasure this relationship.
"They discussed the continuing depth and breadth of the special relationship", a spokesperson said.