The meeting came shortly after First Secretary of State Mr Green said that agreement with the DUP would have to be sealed before finalising the details of the Queen's Speech, setting out the Government's legislative programme.
The Order of the Garter service, attended by the monarch, senior royals and figures from national life, was cancelled to accommodate the June 19 date at short notice.
The Tories and the DUP are locked in discussions about what the programme for Government will be after Mrs May disastrously failed to secure a majority.
There has been no immediate denial of this claim from the Prime Minister's team, leaving Westminster speculating.
May's official spokesman told a media a briefing that the new leader of the House of Commons, Andrea Leadsom, will soon be releasing a statement on the date.
If the Queen were to make her speech next week, then she would likely miss at least one day of racing.
Any hope for a continuation of Tory rule looks set to depend on a consensus with the Northern Irish party, following last week's General Election result in which the Conservatives dramatically lost their majority.
Without the party's support she has no majority in the Commons. "That's certainly something I will emphasis in any contacts that I have with Prime Minister May", he said.
The political turmoil generated by Britain's General Election showed no sign of cooling down and sank the pound even lower in the first session of the week, amid growing doubts over the future of Theresa May's premiership.
People were quick to discuss the delay of the traditional speech, with one tweeting: "Let's spare a thought for the drafters of the Queen's Speech, now starting at an empty sheet of goat skin".
"It is time to stop trying to cling to power and time to admit enough is enough", said Mr Farron, who branded the possible Tory/DUP deal as a "MayDUP government".
He said: "I think we have a job to do and the job is try to provide the most steady government we most possibly can". He told BBC Radio 4's The World at One: "As I'm here today, I'm very firmly of the standpoint the Queen's speech remains on track".
Committee chairman Graham Brady said she was right to take "some responsibility" after calling the snap election and leading "one of the worst campaigns I can recall". That will happen. That will be going on as we speak.
Therefore, as the content of the speech has yet to be confirmed, the Queen's Speech has been delayed.