In the letter, which was sent on Saturday, Ian Blackford and Jo Swinson urge Donald Tusk to give the UK an extension until at least January 31, with no way of leaving early.
The parties hope this would allow enough time for a general election, while also ensuring the UK would not leave the EU without a deal.
The Prime Minister is due to move for a general election on Monday, something both parties have ruled out.
The letter reads: “With an extension to at least January 31 – or later – we, the opposition, can work together to facilitate an election giving the people the chance to decide what the UK’s next steps should be.
SNP leader Mr Blackford said a general election was the only way to end the deadlock in Parliament.
He added: “Boris Johnson’s proposal of an election – but only after he tries to pass his deeply damaging Brexit Bill through Parliament and take the UK out of the EU – is simply not acceptable to us.
“Opposition parties will not be bullied by a Prime Minister who has shown utter contempt for Parliament, and who has attempted to railroad through his damaging deal without a shred of scrutiny or due process.
“On top of that, he is now asking Parliament to back an election without knowing the terms of an extension to the Article 50 process.
“For these reasons we will not support Boris Johnson’s election motion on Monday.
“However, a general election is now the best way to end the current Brexit impasse and that is why the SNP, along with the Liberal Democrats, have written to Donald Tusk seeking an extension to at least 31st January 2020 so that we can remove the risk of a devastating no-deal Brexit.
“If that meaningful extension is secured we will then work together to bring forward an election this year – but on parliament’s terms not on the prime minister’s. This will give people the chance to decide what the UK’s next steps should be.
“This is the only way to unlock what has become a deadlocked parliament and for the UK and the EU to move forward.
“Once this extension is secure we are ready to take the further initiatives that will deliver a general election.”