Lawmakers have discreetly held talks on sweetening the deal for such a swap, as MEPs convene in Strasbourg for just one week every month.
The monthly upheaval costs the bloc £96.34million (€114m) a year, EU auditors say.
Pro-EU President-elect Emmanuel Macron’s victory last week may further help the plans to come to fruition.
Critics have long called for the arrangement to be scrapped but it has stayed in place largely because France would have vetoed any attempt to make the required amendment to the EU treaty.
However, officials say that since Britain’s vote to leave the EU, lawmakers from several groupings in the Parliament have had a change of heart, appearing to now support the idea of a swap.
A text urging to use the “excellent opportunity” of the Brexit-driven transfer of EU agencies from London was backed in April by 75% of EU lawmakers.
The European Medicines Agency (EMA), one of the biggest EU bodies, will have to be moved from London after Britain leaves the EU.
The city also hosts the European Banking Authority, which is likely to relocate to Frankfurt, Germany, beating off competition from Paris, Dublin, Amsterdam among other European cities.
Outgoing French President Francois Hollande had selected Lille as the preferred city to host the drugs regulator.
With pressure increasing on Mr Macron to reverse this choice, it could be a potentially risky move as he prepares for the country’s legislative elections in June.
The EMA’s transfer to France would also require approval from a number of other EU countries, as almost 40 cities have been put forward as potential candidates to host the agency.
According to Politico, senior French centre-right MEP Françoise Grossetête and her colleague Anne Sander called the idea of moving the European Medicines Agency to Strasbourg in exchange for the European Parliament “a dupe market”.
In a strongly worded statement from the European People’s Party group, the two MEPs said the trade-off was “unthinkable.”
Ms Sander said the seat of the Parliament “symbolises the history of Europe, the peace between France and Germany and the reconciliation of our continent”.
She added: “This is clearly not a priority, and France would be ill-advised to give up Strasbourg.”
The two MEPs want Strasbourg reinforced with more EU institutions to shed the EU’s image as a Brussels-based bureaucracy.
They say that Mr Macron has in fact called for a separate parliament for the eurozone, adding that he would want it to be based in Strasbourg.