Varadkar echoed comments made by the EU's Brexit negotiator Michel Barnier on Tuesday that little progress had been made in recent weeks and repeated his own recent warning that there was a real risk of the sides failing to reach a withdrawal treaty by October.
Mr Lidington, a Cabinet Office minister regarded as one of Theresa May's closest allies, told BBC Radio 4's Today programme the document, which will extend to more than 100 pages, would "demonstrate we have thought this through" and show the scale of the UK's ambitions.
Barnier later said that there had been "a little, not very little progress" but warned that more was urgently needed.
There has been "serious criticism" of both proposed models for the UK's customs arrangements after Brexit, a senior minister has acknowledged.
Writing in ConservativeHome, former minister Nick Boles said transitional membership should be extended until March 2022, arguing the political timetable should be adjusted to "reflect the reality on the ground".
She said "substantial progress" needs to be made on the issue of the Irish border by June, saying that the solutions proposed by the United Kingdom government so far would "mean a hard border".
"Certainly the customs partnership, as proposed by the United Kingdom last June, isn't workable, that's the view of the (EU) taskforce and the EU27 and has been rejected, but I do think the customs partnership is closer to being made workable than this proposal of "max fac", Varadkar told parliament.
Under pressure from the European Union to move forward with talks on the future partnership that will follow its exit from the bloc, Britain must first settle on a customs proposal to present to sceptical negotiators in Brussels. Remember, the date of the withdrawal was drawn up by the UK itself - sending this letter in March a year ago, the UK Government has chosen the date of withdrawal in March 2019.
Ms Mogherini said Brexit would be a "loss" to the United Kingdom because it us "giving up the power to use our common foreign and security policy and all its instruments".
"It's simple: Britain is part of Galileo today as an European Union member, but won't be automatically part of Galileo tomorrow as a third-party state", he said.
"The other one is does it affect Northern Ireland in an adverse way and can we mitigate the impact of that?"
Britain's Foreign Secretary Boris Johnson arrives in Downing Street, in London Thomson Reuters LONDON (Reuters) - Britain's foreign minister Boris Johnson said on Monday he had raised concerns with his French counterpart over the European Commission's decision to exclude the country from a new satellite navigation system. We will still share the same region.
The deal could include regular British-EU foreign policy talks, cooperation on military and aid missions, cooperation on defence projects and intelligence sharing, Barnier said.