Donald Trump will discuss how an agreement could be struck after Britain quits the EU when he visits the UK later this week.
But Theresa May's offer to Brussels to share a "common rulebook" on goods and form a new UK-EU free trade area has caused uncertainty over the deal that can be struck with Washington, diplomat Woody Johnson suggested.
He told Anna Foster on BBC Radio 5 live: "The president has stated very clearly that a bilateral with the US is something that he's in favour of.
"He also respects sovereignty. He's talked about sovereignty. He's looking at it in our country as well. So I think the bilateral will be an important discussion that he'll have with the Prime Minister."
He added: "I think that there was a briefing that came out, as I understand it. It was very short, a couple of pages.
"This is a lot more complicated than a couple of pages. I would say that the bilateral agreement, whether we have one or not, is totally up in the air at this point."
Mr Johnson said he "absolutely" believed that a trade deal could be struck.
"From the US standpoint, yes, we'd love to do a bilat, and the president said he'd like to do it quickly, and all hands on deck - so we'll get it done."
President Trump arrives in the UK on Thursday for a visit that will keep him largely away from London.
Mr Johnson said: "It's important just that he be here as well. I think that's going to be very symbolic."