The Liberal Democrats and the Scottish National Party have joined forces in a bid to trigger a 9 December election.
And they have asked EU Council President Donald Tusk to grant a three-month extension of the date for Brexit.
The parties say they reject Boris Johnson's later election date, which they believe would include time for him to "ram through" his Brexit Bill.
But Mr Johnson has said that MPs "cannot hold the country hostage" over Brexit and it was time to "move on".
The prime minister proposed a slightly later date of 12 December for an election, when he made his offer on Thursday.
He said if MPs support his vote in the Commons to call an election, he will use the remaining time before Parliament is dissolved on 6 November to try to pass the legislation for his Brexit deal.
Labour has rejected his election call unless a no-deal Brexit is taken off the table, while the DUP - key allies of the Conservatives until Mr Johnson announced his Brexit deal - has reserved judgment until Monday.
The Lib Dems and SNP say they support an election to "unlock" Parliament - but only on their timetable.
SNP Westminster leader Ian Blackford said he and Lib Dem leader Jo Swinson have also written to Mr Tusk seeking an extension until at least 31 January - a date specified by the UK earlier this month.
He said this was necessary in order to "remove the risk of a devastating no-deal Brexit".
Once that has been secured, he said the two parties would work to bring forward an election "on parliament's terms not on the prime minister's".
"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," Mr Blackford said.
But Mr Johnson warned MPs could "just waste the next three months" debating Brexit, if the UK was given an extension until the end of January.
The PM said he did not want an election and would rather be "getting on" with governing the country but urged MPs to back his election timetable and "move our country forward".
"Millions of businesses and people cannot plan their futures," he said. "This paralysis is causing real damage and the country must move on in 2020."
Gridlock in Parliament
Liberal Democrat leader Jo Swinson said her party could not support Mr Johnson forcing through "a bad deal".
She said that Boris Johnson had "missed his 'do-or-die' deadline (for Brexit of 31 October)".
The Lib Dems and the SNP intend to introduce a short amendment to the Fixed Term Parliaments Act, which would specify the polling day for the next election as 9 December.
It is understood the date was chosen as one of the earliest opportunities for an election, given the time needed to pass the amendment and the required five weeks' notice before polling day.
Ms Swinson said: "We need to get Boris Johnson out of office, unlock the gridlock in Parliament and give people the chance to vote to stay in the EU. "
Mr Johnson's plan for an election would require the agreement of two-thirds of MPs, under the Fixed Term Parliaments Act rules for calling an early general election.
The Lib Dem and SNP amendment to the act would only need a simple majority, if they can get the Parliamentary time for it to be debated.
In their letter to Mr Tusk, Mr Blackford and Ms Swinson urged the EU to grant a longer three-month extension to "give Parliament the assurance that a 'no deal' exit could not take place before that point".
They said they would "work together to facilitate an election giving the people the chance to decide what the UK's next steps should be".
"We believe this is the only way to unlock what has become a deadlocked Parliament and to enable the UK and the EU to move forward - whatever form that takes," they said.
The prime minister has said his preference would be for a short extension until mid- or late-November, but opponents fear this would be used to revive the threat of no-deal to force through his Brexit bill.
What do the Lib Dems and SNP hope to achieve?
Analysis, by BBC political reporter Jessica Parker
The Liberal Democrats and the SNP have a few things in common:
- They think they could do well in an early election.
- They don't like Brexit.
- And they don't want to be seen dancing to Boris Johnson's tune.
So they've composed their own plan for a snap poll on... 9 December.
One source suggested that the ever-so-slightly earlier date could particularly help the Liberal Democrats, because more students would still be in their university towns on the 9th than on 12 December.
But the bigger picture is that these two parties might be worried that if Brussels thinks that Parliament isn't actually going to use a longer extension to do anything meaningful, the EU will grant a shorter one instead... designed to apply pressure on MPs to approve the PM's deal.
So the Lib Dems and the SNP are saying, "give us that longer delay and we will work to break the deadlock".
Their plan, as it stands, hinges on No 10 being interested. Over to Boris Johnson...