The Jerusalem Post (and later others) said that Iran's parliamentary speaker was the one doing negotiations.
He's taken to twitter to say this (in English):
Iranian people demand complete and remorseful punishment of the aggressors. All Irainan officials stand firmly behind their supreme leader and people until this goal is achieved. No negotiations have been held with the US, and fakenews is used to manipulate the financial and oil markets and escape the quagmire in which the US and Israel are trapped.
Now we can take this all at face value or:
1) He's not negotiating directly with the US, as some have claimed but is speaking via Pakistan, Turkey and other intermediaries
2) He's lying because it's unpopular (and dangerous) in Iran to be seen as talking with the US right now
3) Trump is lying
4) The White House is speaking with someone else
In any case, the S&P 500 has trimmed its gains to 1.25% from 2%. In any case, I think the signal here is that the US wants to end the war and that's the gritical detail. Trump does not want to bomb power infrastructure and he wants to get oil flowing again.
On the weekend, it seemed like we were closer than ever to a dangerous escalation but we've stepped back from that. It's also highly likely that the White House is actually talking to someone as it's in everyone's interest to talk. Iranians are also likely cautious as they were attacked twice during negotiations. They're insisting on a ceasefire before they talk. However I imagine the US is insisting on opening the Strait of Hormuz before any talks.
I would emphasize that good headlines don't last long in this market as information travels fast and market demand certainty.