In a column for the Scottish Daily Mail, Mr Sunak claimed Scotland was “central” to his vision for a bright UK future.
He called for an end to the “devolve and forget” mentality as he said it was important to prioritise issues such as the drug death crisis, stressing independence from the UK was the “wrong priority” for Scotland.
Mr Sunak’s comments come as he prepares to hold talks with Scottish Conservative leader Douglas Ross and Tory MSPs in a virtual meeting on Tuesday.
Borders MP John Lamont, who previously supported Penny Mordaunt’s campaign, has now backed Mr Sunak to be the next Conservative PM.
Mr Lamont said the former chancellor “has the experience and plan to take Scotland and the United Kingdom forward together”.
Senior Tory MSPs Miles Briggs, Donald Cameron and Dean Lockhart have also announced they are supporting Mr Sunak.
Mr Sunak said: “We need to stop talking about powers, constitution and process, and we need to all exert our efforts into the real priorities of the Scottish people.
“I am clear that another referendum is the wrong priority at the worst possible moment.
“The SNP are wrong to try and tear the country apart when we should be pulling together.
"Why aren’t they talking about the drugs crisis in Scotland or how we can get more money into Scottish workers’ pockets? Just as I want to be more accountable to people in every corner of the United Kingdom, so too does the SNP need to be more accountable for their responsibilities. They cannot continue to claim credit for all the good things that happen in Scotland, whilst blaming the UK Government for the bad.”
Mr Sunak went on to say he would "stand up to the SNP” to ensure they are “delivering and focusing on the job at hand”.
The leadership contender said the Scottish Tories already hold the SNP to account at Holyrood and he wanted to work closely with them if he wins the leadership contest.
He said: “But as a Conservative UK Government, we can’t just stop a referendum, we also need to drive down support for independence.
"We need to win the argument and show that you can be a patriotic Scot and a proud advocate of our United Kingdom at the same time. One is not exclusive to the other. That’s also why it’s so important that we have an energetic UK Government that is on the side of the Scottish people, delivering on their priorities.
“We have to be a lot more active across the UK, and that means putting an end to the devolve and forget mentality that has shadowed Westminster for decades.”
Mr Sunak said he would use the powers that have come back from Brussels following Brexit to allow him to run “the most active UK Government since devolution began over 20 years ago”.