The pitch from the Prime Minister comes after pro-independence parties, spearheaded by the SNP, won a majority in the Scottish parliament election.
In her victory speech, First Minister Nicola Sturgeon said a Scottish independence referendum was the “will of the country”.
The pronouncement puts the SNP leader on a collision course with Mr Johnson, who has refused to support calls for another referendum, labelling the idea “irresponsible”.
In a letter to Ms Sturgeon and Labour First Minister Mark Drakeford – who saw his party win half of the seats in the Senedd during the Super Thursday elections – the Prime Minister invited the pair to a summit to “discuss our shared challenges”.
The Conservative Party leader said he would be sending a similar letter to the first and deputy first ministers of Northern Ireland.
Congratulating Ms Sturgeon and Mr Drakeford on their re-elections, Mr Johnson said: “I believe passionately that the interests of people across the UK… are best served when we work together.
“I would like to invite you to join me, UK Government colleagues and others at a summit meeting to discuss our shared challenges and how we can work together in the coming months and years to overcome them.
“We will all have our own perspectives and ideas – and we will not always agree – but I am confident that by learning from each other we will be able to build back better, in the interests of the people we serve.”
Mr Johnson said the country needed to “show the same spirit of unity and co-operation that marked our fight against the pandemic” in engineering a Covid-19 recovery for the UK.
But he warned it would be a “difficult journey”, adding: “The broad shoulders of the UK have supported jobs and businesses the length and breadth of the country, but we know that economic recovery will be a serious shared responsibility.”
Both first ministers had their own message for the Downing Street incumbent after their respective election successes.
Following Labour’s election win in Wales and the SNP’s in Scotland, Mr Drakeford called for the Prime Minister to “reset relationships” with the devolved nations.
“This really is a moment that the Prime Minister should seize to reset relationships across the United Kingdom, for a serious examination of the way in which we can create the machinery that will allow us to work together in the future,” he told the PA news agency.
“Not an approach that thinks flying more Union Jacks at the tops of buildings, but proper, respectful relationships that recognise that sovereignty is now dispersed across four parliaments in which we choose to pool it for common purposes.
“That’s the sort of UK that I think will have the very best chance of surviving, because it will be a UK where people want to be here, rather than are instructed to be.”