France and other EU countries have begun planning for Britain crashing out of the EU on March 29 with no deal as negotiations between Brussels and London
appear to be at an impasse.
This would leave a number of key issues in limbo, including the future of Britons living in Europe and EU citizens living in Britain.
The French bill is being considered by the upper house Senate, before it goes to the chamber of deputies.
"The bill is only a first stage in preparing for Brexit," said Jean Bizet, chair of the French Senate's Brexit committee.
Some 300,000 French people live in Britain, while at least 150,000 British citizens reside in France, raising concerns for their future residency status.
The bill will seek to protect French citizens returning from Britain from possible problems over pension rights or having their diplomas recognised in the EU.
It will also take steps to protect the rights of Britons already living in France although France's Europe Minister Natalie Loiseau has made it clear any actions Paris takes will depend on London taking reciprocal steps to guarantee the status of French citizens in the UK.
"We must make sure that in the absence of a deal on March 30, 2019, Britons living in France do not find themselves suddenly with irregular (immigration) status," Loiseau said previously.
The movement of trade is also key, especially at Channel ports and through the Channel tunnel.
Some 30,000 French companies export to Britain, according to the French finance ministry, with goods worth some 31 billion euros ($35.5 billion) in 2017.
France has already begun recruiting extra customs agents to be deployed in Channel ports post-Brexit, and businesses have been warned to prepare for the sudden imposition of trade barriers if the two sides fail to reach a deal.
Ladislas Poniatowski, a conservative overseeing the Senate committee's Brexit report, stressed the government was not being given "a blank cheque" despite the fact it could be allowed to issue emergency decrees.
The committee has already amended the government's original bill, including to add a clause on protecting French and British citizens' employment rights in both countries.