Downing Street has played down the suggestion that Donald Trump could make a formal state visit to the UK in May next year, insisting no date had yet been set for the US president’s long-delayed visit.

Trump’s ambassador to the UK, Woody Johnson, had been asked whether the state visit could be rescheduled to coincide with a commemoration of the end of the second world war. He said: “Between you and me, I think that would be a good time.”

Johnson, who was speaking on BBC Radio 4’s Today programme, added that any agreement for a visit in May 2019 “has to be on both sides”.

But asked about the idea, Theresa May’s spokeswoman said no date had been set. She said: “As you know, we haven’t got a date for you for that. An invitation has been extended for that, and when we have further plans I’m sure we’ll let you know.”

The invitation for a state visit, a formal trip for heads of state involving considerable ceremony and time with the Queen, was extended by May when she became the first overseas leader to visit Trump in the White House after his inauguration.

No date was set, and in mid-2017 it emerged that Trump had told the prime minister he did not want to come to Britain if there were likely to be protests.

The president was then due to travel to London in February this year for a less formal working visit to open the new US embassy, but this was cancelled, again over fears of protests.

Eventually, in July, Trump did visit the UK for a trip in which he was mainly cloistered inside heavily protected official buildings, and then at one of his golf resorts in Scotland. Tens of thousands of people protested against him in London, Manchester, Glasgow, Belfast and elsewhere.

Trump has annoyed Downing Street by arguing that May’s preferred Brexit plan would make it difficult for the UK and US to sign their own rapid trade deal, something reiterated by Johnson in his radio interview.

Johnson said Trump hoped Brexit would “lead to the ability of the US to do a quick, very massive bilateral trade deal that could be the precursor of future trade deals with other countries around the world for Great Britain that will really take you way, way into an exciting future”.

But asked if this would be possible if May’s plan was agreed, he said: “It doesn’t look like it would be possible.”

Asked about the comments, May’s spokeswoman said: “He also recently said that we were the perfect trading partner for the US. Both sides have been clear through the process, ourselves and the US, that we want an ambitious trade agreement, and we stand ready to conclude such an agreement as a priority after we leave the European Union.”