Washington (CNN) French President Emmanuel Macron arrived in Washington hoping to convince President Donald Trump to stick with the Iran nuclear deal.

Instead, Macron now believes it's going to be necessary, following his discussions with Trump, to negotiate an additional, new deal with Iran -- and the US would like it to be even more formal, possibly a legally-binding treaty.

According to sources familiar with the discussions between Trump and the French President, the two hashed out a plan to begin negotiating another deal with the Iranians right away, one that will sit alongside and go further than the 2015 agreement that constrains Tehran's nuclear program in exchange for sanctions relief.

The "treaty" would seek to put multiple outstanding issues that Trump has railed against into one box: Iran's ballistic missile program, the "sunset clauses" that ease limits on Iran's centrifuges and nuclear enrichment under the current deal, and even Iran's behavior in regional conflicts like the ones in Yemen and Syria.

This goes well beyond the current discussions that have continued for months between the US and Europe, with European allies lately growing more optimistic that they could convince the US to reach a "political agreement" to stay in the nuclear deal, and then work together over time with Iran to address what happens once the deal expires.

Read More