Tusk's remark comes after British Prime Minister Boris Johnson and Varadkar held "detailed and constructive" talks for more than two hours on Thursday, with the latter saying the "very positive" meeting meant negotiations could resume in Brussels, the BBC reported.
Following the meeting, Varadkar told reporters that the talks were at a "very sensitive stage" but he was now "convinced" the UK wanted a deal, adding: "I do see a pathway towards an agreement in the coming weeks."
Speaking on a visit to Cyprus, Tusk said on Friday that the UK had not "come forward with a workable, realistic proposal", but added he had received "positive signals" from Varadkar that a deal was still possible.
"Of course there is no guarantee of success and time is practically up but even the slightest chance must be used.
"A no-deal Brexit will never be the choice of the EU," he added.
However, there were, he said, still issues over "consent and democracy" and ensuring there is no customs border between Northern Ireland and the Republic of Ireland.
Tusk's comments also comes on the same day that UK's Brexit Secretary Stephen Barclay is slated to meet the European Union's (EU) chief negotiator Michel Barnier in Brussels.
Barclay and Barnier's talks come ahead of a crunch EU summit on October 17 and 18, which is seen as the last chance for the UK and the EU to agree a deal ahead of the Brexit deadline, the BBC reported.
The UK is due to leave the EU at 11 p.m. (local time) on October 31.
With three weeks to go until the Brexit deadline, Johnson continues to insist the UK will leave the EU with or without a deal at the end of the month.
That is despite the so-called Benn Act - passed by MPs last month - demanding he request a delay to the Article 50 deadline from the EU until January 2020 if a deal has not been agreed before October 19.
Should any new deal be reached between the EU and UK, it will still have to be approved by MPs in Parliament.
( With inputs from IANS )