Prime Minister Boris Johnson has invited the leaders of the devolved nations to a summit on how "Team UK" can recover from the pandemic.
It follows the SNP's fourth Scottish Parliament election win in a row and Labour's win in the Welsh Parliament.
The SNP's Nicola Sturgeon said there was "no democratic justification" for blocking an independence vote.
But Mr Johnson said talk of "ripping our country apart" would be "irresponsible and reckless".
He has telephoned Welsh First Minister Mark Drakeford and Ms Sturgeon, who has confirmed she will attend the summit.
An SNP spokesman said Scotland's First Minister had "re-iterated her intention to ensure that the people of Scotland can choose our own future when the crisis is over, and made clear that the question of a referendum is now a matter of when - not if".
In letters sent to the leaders, the prime minister highlighted the Covid vaccine rollout as an example of "Team UK in action" and called for a continued "spirit of unity and cooperation".
The SNP fell one seat short of an overall majority in the Scottish Parliament elections, but with eight Scottish Greens elected, Holyrood will have a pro-independence majority.
Scotland held a referendum on leaving the in 2014 - billed as a "once-in-a-generation" event - with just over 55% voting to remain part of the UK.
But after the UK backed leaving the EU in 2016, Ms Sturgeon began pushing for another referendum on Scottish independence.
She told the BBC's Andrew Marr Show she did not believe any bid for a second referendum would end up in a legal challenge from the UK government.
She also warned that any such move it would mean Westminster "saying the UK is no longer a union based on consent".
Cabinet Office Minister Michael Gove said the SNP could not expect a second referendum to be approved as the party had not secured a majority in the Scottish Parliament.
He said there was a "significant difference" between Ms Sturgeon's win and her predecessor Alex Salmond's majority in 2011.
Asked whether the UK government would try to block legislation from the Scottish Parliament on a second referendum, Mr Gove said: "We're not going near there."
Mr Drakeford, who is not pushing for more powers for the Welsh Parliament, urged the prime minister to "reset relationships" with the devolved administrations.
Northern Ireland did not have elections on Thursday, but Mr Johnson confirmed its first minister and deputy first minister would also be invited to the summit.
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