Sources said ministers planned to lay the groundwork for a new Commons vote.
The government plans to set out its achievements in Iraq so far with RAF air strikes and training of Kurdish and Iraqi security forces, sources said.
The BBC also understands a small force could be sent to Libya.
The team of about 20 troops would be sent to help Libya secure its borders - which is seen as a crucial step towards stemming the flows of migrants into Europe.
An MoD spokesperson commented: "The UK, along with international partners, is supporting the process to form a recognised Libyan government and we are developing plans to provide support once this is done; it is too early to discuss the exact nature of this."
According to sources, ministers intend to argue that the mission to defeat IS now has to turn to Syria, and that the UK should play a role in that effort.
Describing the government's plans to increase its military involvement in the region, a source said: "The government is essentially posing a question: Could we do more? Should we do more? But Syria is where the fight should be taken to."
Another source added: "The government will say the campaign in Iraq has been a success. IS has been degraded, land has been taken back. Some of their leaders have been killed. But the problem is across the border in Syria."
Two years ago MPs rejected possible UK military action against Syrian President Bashar al-Assad's government.
A new parliamentary vote on the issue is not imminent, but the case will continue to be made throughout September in the run-up to the anniversary of the launch of UK airstrikes in Iraq, sources added.
They said any move to extend UK military action to Syria would only involve an extension of airstrikes against IS targets and involvement in coalition special forces operations.
The timing of the separate, small deployment to Libya is expected to depend on when Libya can form a unity government, and may be part of a broader effort. Nato is poised to go back into Libya to rebuild the country's defence and military once there is more political stability.
Nick Hopkins will have more on this story on Newsnight on Friday at 22:30 BST on BBC Two.
Ministers will start to make a case for British military action in Syria next week - with Downing Street keen to take the "next step" against so-called Islamic State - the BBC understands.