The prime minister told MPs the plan must be treated seriously, but also "tested out properly" to ensure it was not a "delaying tactic" or a "ruse".
He said the resolution would include "a proper timetable, process and consequences if it's not done".
The plan is for stockpiles to go under international control and be destroyed.
Mr Cameron said the Russian plan, apparently welcomed by Syria, was an idea "definitely worth exploring" but added that he entered the process with a "hard head and cool calculations".
Mr Cameron, who spoke to US President Obama and France's President Hollande on Tuesday, said he wanted to ensure that this was not just a way "to simply buy time" for Syrian President Bashar al-Assad.
Mr Cameron told the liaison committee of senior MPs that if one of the world's largest chemical weapons stockpiles could be eliminated it would be a "significant step forward".
But he said the UK and its allies must remain sceptical.
The wording of a joint US, French and UK resolution on Syria's chemical weapons is unlikely to be agreed on Tuesday, Whitehall sources have told the BBC.
By Nick RobinsonPolitical editor
I am told that the wording is still very much a work in progress
Diplomats from the three allies are said to be discussing the questions of "what, where, when, who and how" - in other words what weapons should be removed from Syria, where should they be taken to, according to what timetable and who should supervise it, BBC Political Editor Nick Robinson said.
There was also a discussion about what type of UN resolution should be tabled - whether it should be a so-called chapter 7 or chapter 6 resolution, he added.
Chapter 7 resolutions are binding, seen as authorising military action if other measures do not succeed. Chapter 6 resolutions, in contrast, are entitled "Pacific Settlement of Disputes" and stipulate that parties to a dispute should use peaceful methods of resolving disputes, such as negotiation and mediation.
Russia's ambassador to France, Alexandre Orlov, earlier said Moscow was ready to negotiate a UN resolution that foresees international control of Syria's chemical weapons, but ruled out a recourse to use force to impose it.
President Obama said he would put plans for a US military strike on hold if Syria agreed to place its stockpiles under international control.
He described the idea - in response to the Syrian government's suspected use of chemical weapons on civilians - as a "potential breakthrough", but said he was sceptical the Syrian government would follow through.
Russian Foreign Minister Sergei Lavrov said his Syrian counterpart, Walid Muallem, welcomed his country's proposal.
The US accuses Damascus of war crimes including the use of chemical weapons, allegations denied by the regime.
The crisis, which began when anti-government protests were brutally suppressed, has left about 100,000 dead and forced more than two million Syrians to flee the country.
The suspected chemical weapons attack in Damascus on 21 August killed more than 1,400, according to US authorities, entrenching divisions among world leaders on how to respond.
The US has said that the use of chemical weapons crosses a "global red line", and President Obama is seeking approval from Congress for military action in response to the attack.
He gave a series of television interviews on Monday aimed at building support in Congress, as he maintained a limited strike was needed to punish President Assad's regime.
Key Syria ally Russia has been highly critical of plans to intervene, arguing that there is no proof the Syrian government was behind the attack, and warning of "catastrophic consequences" of military intervention.
While Mr Cameron conceded there was "never going to be unanimity" on Syria at last week's summit in St Petersburg, those who supported a strong response following chemical weapons attacks on civilians by the Assad regime made an "extremely powerful" case, he said.
"I am clear that it was right to advocate a strong response to the indiscriminate gassing of men, women and children in Syria and to make that case here in the chamber," the PM said.
He said he understood and respected the House of Commons' recent decision to vote against intervention, so the UK would not be part of any military action but would continue to press for the strongest possible response, including at the UN.
But shadow chancellor Ed Balls told Channel 4 news that "jaws dropped" when Mr Cameron ruled out UK military action in Syria.
Mr Balls insisted Labour would be prepared to revisit the issue if Mr Cameron were able to present new evidence to MPs.
"But he is the prime minister - he has got to lead," the shadow chancellor said.
Meanwhile, Business Secretary Vince Cable was accused of allowing the export to Syria of chemicals which could be used in the manufacture of chemical weapons, in breach of the government's own policy.
Sir John Stanley, the chairman of the Commons Committee on Arms Export Controls, said the export licences, issued in January 2012, should never have been approved.
He called on Mr Cable to release the name of the company involved, declaring that he could no longer hide behind the need for commercial confidentiality.
In correspondence with the committee, Mr Cable had said the licences for the export of sodium fluoride and potassium fluoride were granted after an assessment in relation to both EU and UK rules - including whether there was a "clear risk that they might be used for internal repression" - and concluded there were no grounds for refusal.
David Cameron has said the UK is to table a UN resolution with France and the US about the Russian plan for Syria to hand over its chemical weapons.