With at least four other Conservatives likely to back the amendment, it could pass if Labour supported the measure.
Responding to Mr Tusk's remarks, British sources told Sky News that Mr Tusk should probably wait until the United Kingdom position is set out properly before passing judgement.
"You've got to remember we are leaving the European Union on March 29 (2019), we will be out of the treaties on that day, we will not have any say in the rules that are made and therefore people who come after that day ought not to be allowed to have the full and permanent free movement rights".
New figures have revealed that 45 per cent of Leave voters believe the BBC News has an anti-Brexit bias, compared to 14 per cent of Remain voters.
"It will result in #Brexit in name, but not in reality", he wrote on Twitter.
May's chosen approach rejects the idea of being in any form of customs unions with the EU, with the health secretary, Jeremy Hunt, stressing on Friday that this would not happen.
Senior ministers suggested agreement had been reached - following their marathon talks at the Prime Minister's country retreat - on the UK's desired outcome for Brexit; which would see close future alignment to European Union regulations in some areas, but no alignment in others.
Britain's prime minister is also feeling the heat from Brexit hardliners in her party who have called her acceptance of a status-quo transition after Britain leaves in March 2019 everything from a "betrayal" to "a perversion of democracy".
Mr Gualtieri, who is the Chair of the Committee on Economic and Monetary Affairs, said the Prime Minister's speech next week will allow the European Union to outline their Brexit guidelines.
"But that must always be on a voluntary basis", he told Sky News.
The biggest response came from the 40 per cent that thought the Mail was pro-Brexit, with 37 per cent saying the same for The Sun and 31 per cent for the Express - all three papers campaigned for Britain to exit the EU.
"You've got to remember we are leaving the European Union on March 29 (2019), we will be out of the treaties on that day, we will not have any say in the rules that are made and therefore people who come after that day ought not to be allowed to have the full and permanent free movement rights". Juncker made the comment at an European Union summit when asked about a highlevel British cabinet meeting chaired by premier Theresa May aimed at thrashing out a plan for a future partnership with the EU.
The EU position states that the British views on regulatory issues in future relations, including the tripartite approach, are not consistent with the principles in the EU guidelines.