A plan is being hatched in the European Parliament to make French President-elect Emmanuel Macron an offer he could very well refuse: take the European Medicines Agency to Strasbourg, and in exchange let the European Parliament meet only in Brussels.

An influential member of the Parliament said the idea has been floating around in recent weeks as a way to stop the institution from spending some €114 million a year on traveling once a month to Strasbourg for only a few days. Another bonus: The Louise Weiss Parliament building could become the new office space for the drug regulator's nearly 900 employees.

The race to host the EMA post Brexit, when EU officials say it must leave its London headquarters, is fierce, with a majority of EU countries in the running to get it. EU leaders are expected to decide the new host city in October.

London is also the current home of the European Banking Authority, and the idea is to sweeten the pot by offering Macron, a former economy minister, that smaller agency as well.

The possibility was vaguely floated in the Parliament in late April when MEPs discussed the EU budget. "The withdrawal of the U.K. and the need to relocate the European agencies, which currently have their seats in the U.K., could provide an excellent opportunity to solve several issues at the same time," according to the Parliament at the time. MEPs are now expected to make a more concrete decision on the EMA-Parliament swap before it goes into summer recess.

The MEP, who is promoting the plan in the Parliament and among national governments, believes the EMA is something permanent that could be offered to the city of Strasbourg in exchange for giving up on the nomad European Parliament. For MEPs and their staff, it means cutting unnecessary travel and wasted money and time, while to the French city, it would bring permanent jobs and guests for its hotels and restaurants.

The EMA would use the Parliament's building in Strasbourg permanently, and not just for three or four days a month, as happens now, the MEP said. Strasbourg also "offers an environment with numerous medical experts, such as the high-performing medical faculty at the Strasbourg University, a pharmaceutical faculty, and different research institutions," the MEP said.

The downside? Strasbourg local authorities and the French government have to be convinced to accept such an idea as the country heads into parliamentary elections next month.

The potential loss of the prestigious Strasbourg seat for the European Parliament and the money that comes with it could be used in the campaign rhetoric of Marine Le Pen's National Front against Macron's movement, the MEP worries.

If the plan survives and wins agreement, the EU treaties that designate Strasbourg as the official seat of the European Parliament would have to be changed. To get there, all EU countries now bidding heavily for the EMA would have to agree to send it to Strasbourg.

However, senior French center-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 Thursday, the two MEPs said the trade-off was "unthinkable."

The seat of the Parliament "symbolizes the history of Europe, the peace between France and Germany and the reconciliation of our continent, " Sander said.

Sander 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. Macron has called for a separate parliament for the eurozone, and wants it to be based in Strasbourg, they said.

Giulia Paravicini, Sarah Wheaton and Quentin Ariès contributed reporting.