The Scottish Labour leader demanded the two administrations be forced to work together as he insisted his party “champions devolution”.
In a speech hosted by the Fabian Society at Westminster on Monday, Mr Sarwar said “co-operation and not conflict” must be at the heart of government as he accused the Conservatives and the SNP of stoking “bitterness, division and anger” for their own political gain.
The Glasgow MSP has proposed new joint governance councils that would “bring together the leaders of the UK and the nations on an equal footing” and provide a forum for dispute resolution.
Mr Sarwar also accused Nicola Sturgeon of pitting Scots against each other through her “obsession” with independence, but did hint at a softening of Labour’s position.
He said: "The Scottish people are sovereign and have the right to determine the best form of government suited to our needs.”
Outlining Scottish Labour’s proposals, Mr Sarwar insisted the legal duty to co-operate was not just a “political wheeze”.
He said: “This would require joint working between governments in areas of shared interest
“Secondly, we are proposing new joint governance councils – or whatever we ultimately decide to call them.
“They would be designed to heal the bad relationship that exists today and provide a constructive forum for dispute resolution
“Too often, the current UK Government keeps the Scottish Government in the dark. And too often the current Scottish Government deliberately seeks disagreement with the UK Government.
“This does not lead to good governance – it undermines the Union. We need these new rules of engagement to heal it.”
Mr Sarwar vowed his party would bring “new hope” to politics as he stressed: “Labour will do no deal with the SNP – no deal, no pact, no behind-closed-doors arrangement, no coalition.”
He also said a Labour government would abolish the House of Lords – a move originally contained in the party's 2019 manifesto – with the institution to be replaced by a new “Senate of the Nations and Regions”.
He said: “The House of Lords must be abolished and replaced with an institution which better reflects the make-up and the identity of the United Kingdom.
“We shouldn’t forget that it is Labour Lords who have stopped some of the worst excesses of this Tory government.
“But we need something that is more reflective of modern Britain, and which gives Scotland and other parts of the UK a greater say in UK-wide legislation.”
The new senate would have directly-elected members, with a mandate to represent their nation or region, better reflect the diversity of modern Britain and give Scotland a greater say in UK-wide laws.
Asked about details of the new senate, Mr Sarwar said: “I’m less concerned about the location, I’m less concerned about the make-up. I’m more concerned right now with the principle, the fact that we need to abolish an arcane institution, the principle that we need a more outright democratic alternative, that it needs to be reflective of the nations and the regions”.
Mr Sarwar also accused the First Minister of pursuing independence against the wishes of Scotland.
He said: “The truth is the pandemic Nicola we saw on our TV screens during the pandemic and who promised to pull us through has gone.
“She has been replaced by the partisan Nicola Sturgeon, who wants to pit Scot against Scot for her own obsession.
“Worse still she has taken the ‘thank you’ she was given from the public last year, and the promise she made to lead us through the recovery, to instead pursue a referendum the vast majority of Scots do not want.
“I appreciate that for people in London and across the rest of the UK, another referendum seems such a strange priority for anyone in government when we face the greatest cost-of-living crisis of modern times.
“And it is the case that the majority of Scots believe that too. The SNP is trying to drag people back into the arguments of the past because it has no ideas for the future.”
The Scottish Labour leader also warned Ms Sturgeon her party could either support a Labour Government or enable the Tories.
Mr Sarwar said: “Boris Johnson is a gift for the SNP.
“Every time they fail, they either shout ‘independence’, or they point at Boris Johnson’s Tories and say ‘at least we’re not as bad as that lot over there’.
“Another Tory government is precisely what the SNP wants too, because it allows them to continue their grievance campaign.
“For more than a decade, the SNP and the Tories have stoked up bitterness, division and anger and created a politics of us versus them.
"At the next election, we will be fighting for every vote and we are aiming to form a majority Labour Government.
“Should we fall short of that, and be in a position to form a minority government, the SNP will face a simple choice.
“It can choose to keep the Tories in power, or choose to back a Labour government.
“And I dare Nicola Sturgeon to back the Tories and put them back in power, and see how Scotland responds.”
The SNP compared Labour to the Tories, pointing to Sir Keir Starmer’s speech on Monday evening entitled “Making Brexit Work”.
SNP MSP Kaukab Stewart said: “It’s staggering that Labour are now trying to outdo the Tories on their support for Brexit, and their Trump-like attempt to deny the results of democratic elections in Scotland is embarrassing.
“The only way to protect Scotland from the damage of a disastrous Brexit we didn’t vote for is with the full powers of independence – and Anas Sarwar’s absurd suggestion that all Scotland needs is a reformed House of Lords shows just how out of touch he is and only strengthens the case for independence still further.
“Nobody can take Labour seriously when their leader rules out working with the SNP in a progressive alliance to kick the Tories out and signs up to grubby deals with the Tories in councils across Scotland this year.”
Scottish Conservative chairman Craig Hoy MSP said: “Anas Sarwar issued – and then broke – the same promise before, so pro-union voters know they can’t trust him to keep to his word.
“In the run up to May’s council elections, he could not have been clearer, repeatedly, that Labour would do no deals with the SNP. But no sooner had the votes been counted than Labour happily teamed up with the SNP in Dumfries and Galloway, rather than work with the biggest party on the council – the Scottish Conservatives – to lock out the Nationalists.
“Anas Sarwar may try to gloss over this inconvenient truth, but pro-union voters know that only the Scottish Conservatives can be trusted to stand up to the SNP.”
Lib Dems Scottish affairs spokesperson Wendy Chamberlain instead offered to work with Labour to make a stronger UK.
She said: "Labour may be a little late to the conversation over how to build a better UK, but it is better late than never.
"Throughout the debates over independence and Brexit, the Liberal Democrats have made the case for a new and lasting dispute resolution mechanism that puts the needs of our constituents ahead of petty party politicking by the Conservatives and SNP. Likewise reform of the House of Lords has long been a liberal goal.
"Though it does not go far enough in terms of support for building a federal UK and delivering fair votes, this paper contains some worthwhile ideas.
"My party stands ready to take part in constructive discussions with anyone who is serious about making the country work better."