He said countries that support Assad's regime "have to ask themselves some hard questions".
Italian Foreign Minister Angelino Alfano said Sunday that Europe's broad support for the US military strikes had contributed to a "renewed harmony" between the United States and its partners.
A Downing Street spokesman said: "Theresa May spoke with US President Trump to discuss last week's chemical weapons attack in Syria and the US response".
Japan's Foreign Minister Fumio Kishida said he told Tillerson that Tokyo supports the U.S. in its push to "deter the spread and use of chemical weapons", and discussed the pressing North Korean nuclear threat.
President Donald Trump on Thursday approved a military strike against a Syrian air field in the wake of an apparent chemical weapons attack in Syria that killed dozens of civilians.
The poll found that 87 percent of Americans believe that the Syrian government's use of chemical weapons is immoral, and 45 percent think it is a direct threat to the U.S.
The ministers will also talk about growing tensions with North Korea, with the United States moving a navy strike group near the Korean peninsula amid concerns over Pyongyang's nuclear ambitions.
Syria, though, topped the agenda.
The two leaders spoke about the crisis by telephone as the U.S. president thanked the PM for her backing of his military strikes against Syrian government forces last week.
"There's no doubt the USA action is a game-changer in Syria". With stability restored, they say, conditions will be ripe for a United Nations -brokered political transition.
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US officials say the chemical weapons were stored there and that those elements add to the conclusion that Russian Federation was complicit in the attack.
There are sanctions in place over Russia's behaviour in Ukraine and the Crimea, but they are tightly focused on individuals within the Russian regime.
A Government source told me the UK's focus is on defeating so-called Islamic State first - a job they think is almost done in Iraq and could be completed in Syria in about six months.
The British foreign secretary had been due to visit Russian counterpart Sergey Lavrov in Moscow before the G-7 meeting. As he laid a wreath, he alluded to the Syria chemical attack.
"We will be discussing the possibility of further sanctions certainly on some of the Syrian military figures and indeed on some of the Russian military figures who have been involved in co-ordinating the Syrian military efforts and are thereby contaminated by the appalling behaviour of the Assad regime", Mr Johnson told reporters.
Washington has sent mixed signals about whether it shares the determination of allies such as Britain that Assad must be removed from power.
"Returning to pseudo-attempts to resolve the crisis by repeating mantras that Assad must step down can not help sort things out", Dmitry Peskov, spokesman for Russian President Vladimir Putin, said on Monday.
Among European nations, there are also differences.
"The United States should first tell Russian Federation that this kind of a war crime is unacceptable in the world today". "To prevent military violence to escalate on and on, it's all about this".
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