The warning came ahead of a meeting between the pair at the G7 summit that took off in Biarritz, France, today.
The PM was branded ‘Mr no-deal’ by Tusk, who is president of the EU’s European Council.
Before Johnson’s arrival, Tusk said he was willing to listen to ‘realistic ideas’ but was not willing to compromise with the UK if we crash out of the European Union without a deal.
The former Polish Prime Minister said: ‘He will be the third British Conservative prime minister with whom I will discuss Brexit.
‘The EU was always open to co-operation when David Cameron wanted to avoid Brexit, when Theresa May wanted to avoid a no-deal Brexit, and we will also be ready now to hold serious talks with Prime Minister Johnson.’
However he also warned: ‘One thing I will not co-operate on is no-deal. I still hope that Prime Minster Johnson will not like to go down in history as ‘Mr no-deal’.
‘We are willing to listen to ideas that are operational, realistic and acceptable to all member states including Ireland, if and when the UK Government is ready to put them on the table.’
Boris Johnson has vowed to leave the EU ‘deal or no deal’ on October 31.
German Chancellor Angela Merkel has given him 30 days to come up with a solution to the Irish Backstop – a major sticking point in the Brexit debate.
What is the Irish Backstop?
The Irish Backstop is a clause in the Brexit Withdrawal Agreement to maintain a seamless border between Northern Ireland and the Republic of Ireland when the UK leaves the EU.
It involves the UK remaining in a customs union until a post-Brexit trade deal is struck with the European Union. In addition, Northern Ireland would remain aligned to parts of the European single market.
Hard-line Brexiteers oppose the clause because it would involve the UK retaining a very close relationship with the EU for an indefinite period if no solution has been found for the Irish border.
However, the EU insist that any Brexit deal must contain a backstop.
President of France Emmanuel Macron backed her calls for a new resolution, but said it would not be possible to strike a ‘substantially different’ deal to the one agreed under former PM Theresa May.
Johnson met with the leaders of France and Germany earlier this week in his bid to renegotiate the UK’s current withdrawal agreement.
Building on those talks, he is expected to lay down his plans for Brexit in a meeting with Tusk later today.
Before setting off for the coastal resort of Biarritz, the PM warned Brexit critics they are ‘gravely mistaken’ about the UK losing its place on the world stage in the event of a no-deal.
The summit will last until Monday and is Johnson’s first G7 meeting as Prime Minister.
He is set to meet President Donald Trump tomorrow for their first face-to-face discussion since he entered Downing Street.
Ahead of the summit, Mr Johnson said: ‘Some people question the democratic decision this country has made, fearing that we will retreat from the world. Some think Britain’s best days are behind us.
‘To those people I say: you are gravely mistaken.’
Mr Johnson will meet the US president – who has repeatedly praised his leadership qualities – on Sunday morning.
His comments may be seen as a rebuke to Emmanuel Macron, who suggested that a post-Brexit trade deal with the US could see Britain as a ‘junior partner’ in a position of ‘historic vassalisation’.