Antti Rinne said he and French President Emmanuel Macron agreed the UK needed to produce the proposals in writing by the end of September, adding if not, "then it's over".
Finland currently holds the EU's rotating presidency.
A Downing Street source said: "We will continue negotiating and put forward proposals at the appropriate time."
Mr Johnson has said a deal is possible at a crucial summit of EU leaders on 17 October, but he has insisted Brexit will happen by the 31 October deadline, even if a deal is not agreed.
The UK government said talks with the EU have been making progress since Mr Johnson came into No 10 in July.
It said it had put forward "a number of proposals" as alternatives to the Irish border backstop - the policy aimed at preventing the return of a hard border on the island of Ireland and a key sticking point in former PM Theresa May's Brexit deal.
But Mr Johnson has repeatedly refused to reveal details of the proposals in interviews, saying he did not want to negotiate in public.
The EU has continued to criticise the UK for not putting any plans in writing.
Earlier, the European Commission President, Jean-Claude Juncker, said a meeting with Mr Johnson on Monday had been "constructive".
But he said until proposals had been put forward, "I will not be able to tell you, looking you straight in the eye, that any real progress has been achieved".
Mr Rinne spoke to reporters after a meeting with the French president in Paris on Wednesday.
He said: "We both agreed that it is now time for Boris Johnson to produce his own proposals in writing - if they exist.
"If no proposals are received by the end of September, then it's over."
The Finnish PM intends to discuss the new deadline with the President of the European Council, Donald Tusk, and Mr Johnson in the coming days, but the position has not yet been agreed with other EU nations.
An official at the Elysee said the plan was "not at all a new proposal" and added: "If we don't get the proposals before the end of September, we will not have enough time to discuss them before the summit in October."