At the GQ awards on Tuesday September 6, they turned up to the event together, walking the carpet arm in arm.
Overall, he sounds confident, both of the film and his vision of it, which he says will tip a hat to the groundbreaking designs of Ken Adam in the first Bond films.
Several times he mentions what he believes will be at the heart of the new film: character.
But what do they have to say about the alleged affair?
Here’s what you need to know about how their rumoured romance...
Despite all the flirting, divorcee Rebecca has denied finding love again in the silver fox.