Writing to Mr Tusk, the PM confirmed he is committed to ensuring there is no return to a hard border on the island of Ireland.
After more than three years of Brexit crisis, the United Kingdom is heading toward a showdown with the European Union as Johnson has vowed to leave the bloc on October 31 without a deal unless it agrees to renegotiate the divorce terms.
Karendeep Kaur, senior immigration consultant at Migrate UK, said prime minister Boris Johnson's unwillingness to phase out the current immigration rules have not allowed for European Union nationals now in the UK, or those looking to enter before 2021, to organise their affairs.
It comes as a senior US politician warned that legislators would move to block a future US-UK trade deal if it puts the Good Friday Agreement at risk by introducing a hard border between Northern Ireland and the Republic of Ireland.
"The scenario that is now becoming the most likely is the no-deal scenario", the official said.
A no deal Brexit is starting to look inevitable. "Even if they do not recognize it".
Responding to a letter sent by Johnson to European officials, Commissioner Hogan said the EU shares the prime minister's commitment to an orderly Brexit and upholding the Good Friday Agreement, but regretted the lack of a forthcoming alternative from the United Kingdom to replace the backstop.
The official, speaking on condition of anonymity, rejected Johnson's demand that the so-called "backstop" mechanism to avoid border checks between Ireland and Northern Ireland be scrapped.
European diplomats expect little progress on Brexit until the British domestic landscape becomes clearer when parliament returns on September 3 - after which the opposition Labour Party has vowed to try to collapse Johnson's government.
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He says this will free up staff "to get on with preparing for our departure on October 31 and seizing the opportunities that lie ahead".
Under the plan Ireland would gain a "special dispensation" from Brussels to leave its trading rules until new "alternative arrangements" to the Northern Ireland backstop are established.
Merkel struck a cautiously hopeful note on Tuesday, declaring that the European Union was open to "a practical arrangement" for the Irish border if it ensured trade and peace under the Good Friday Agreement.
On the agenda is a meeting between German Chancellor Angela Merkel and Prime Minister Boris Johnson later in the day; we are told by one analyst that any positive headlines here could see Sterling garner further support.
Johnson has called the Irish backstop "unviable", insisting on the long-held belief held by pro-Leave legislators of the Conservative party that unspecified technologically "creative" solutions can replace the need for a border, despite the fact that there is no global precedent to that effect. He will be visiting German Chancellor Merkel and French President Macron this week.
Tabloid-style Bild daily nominated Johnson as its "loser of the day" Wednesday after he "hit a brick wall" in his attempts to convince Merkel and Tusk to renegotiate parts of the withdrawal agreement. But if Northern Ireland and Ireland have different regulations and tax policies, diplomats say, some form of border controls will be necessary to prevent smuggling.
Johnson and Irish Prime Minister Leo Varadkar spoke for almost an hour on Monday, as Varadkar emphasized that the need for a "legally operable guarantee" to prevent a hard border.
"We are not in the business of being steam-rolled at the end of this because a British prime minister has rolled out new red lines", he said.