David Cameron has said he plans to stage a quick Commons vote on extending airstrikes in Syria next week, saying he will make an oral statement to MPs this Thursday and then ask them to consider his proposals over the weekend before going to a full debate and vote.

The prime minister told MPs on Monday: “I don’t want to bounce the house into this. Members of parliament will be able to take it away, consider it over the weekend, and then we go to having a full day’s debate and proper consideration, and a vote. We shouldn’t take too long over it. Every day that we spend is a day that we’re not getting to grips with the Isil menace.”

Cameron’s timetable also gives time for the defence secretary, Michael Fallon, to brief individual MPs. Many Labour MPs and probably a majority of the shadow cabinet are open to airstrikes so long as they are part of a wider political and diplomatic strategy, including ideally a ceasefire between Syria’s president, Bashar al-Assad, and most rebel forces apart from Isis (also known as Isil) and al-Nusra front.

Cameron, who visited the site of terrorist attacks in Paris on Monday morning, told MPs that the attacks in Paris showed Isis was not a remote threat thousands of miles away, and it was his firm conviction that Britain should join the air campaign in Syria.

He said: “My aim here is to bring together the biggest possible majority across this house for taking the action that I think is necessary. And I’m not saying that we will solve this problem simply by crossing a line from Iraq into Syria.

“We’ll solve this problem if we have a political strategy, a diplomatic strategy, a humanitarian strategy.”

The US secretary of state, John Kerry, said such a ceasefire was possible within a fortnight, although many are sceptical that Russia or Iran would be willing to abandon Assad in the short term. There are also still doubts that Russia has switched its bombing campaign to attacks on Isis, as opposed to defending Assad from rebel forces.

Britain and the US insist he must stand down following elections within 18 months of all-party talks on a new constitution. Iran and Russia vowed on Monday to oppose external efforts to oust him.

Cameron probably needs as many as 40 or 50 Labour MPs to back airstrikes to ensure he has a clear Commons majority. His biggest domestic political problem is to reassure MPs that he knows the identity and the loyalties of any forces on the ground that would seize ground from Isis with the help of an air campaign.

The Labour leader, Jeremy Corbyn, is instinctively opposed to airstrikes but is unsure whether to impose collective discipline on his party. His Labour support group Momentum is already campaigning to press he party’s MPs to oppose strikes.

But writing in the Guardian, Dan Jarvis, a former officer in the Parachute Regiment and shadow foreign office minister, gave a conditional signal that he would back airstrikes. He said the UN security council resolution passed last week gives Britain a compelling political and legal mandate to act, so long as the prime minister showed he had a wider strategy to secure peace.

Jarvis says this would require troops on the ground led by regional forces since the presence of British troops would only reinforce the errors of the Iraq war.

He adds that a viable plan would have to address the question of Assad and his brutal regime. “No adequate long-term solution can have any place for a dictator who has used chemical weapons against his own people. We need to build upon the progress made between countries at the Vienna conference and work towards a transition where he goes once and for all.”

Jarvis, who is likely to be a hugely influential voice in the Syria debate on the Labour benches, says the party must learn lessons from the past such as Iraq, but adds: “We must not become prisoners of it either.

“This is a moment when we should put party politics aside in the national interest of our country. We have a duty to stand together and confront this common enemy as one. If the prime minister can show he has a wider strategy to do that, then he will have my support.”

Tory sceptics, such as Dr Julian Lewis, the chair of the defence select committee, argued that it was not possible to defeat Isis without troops on the ground, and there were no credible moderate ground forces capable of defeating Isis, thus requiring the west to accept Assad will have to remain in power in the medium term.

But Fallon said: “There are moderate forces fighting Daesh [another name for Isis] in Syria that have also been engaged in the civil war. The key is to bring the civil war to an end as quickly as possible so that we can focus on dealing with Daesh.”