Ruth Davidson has reportedly urged some party members to “take a long, hard look at themselves” after a poll suggested a majority of Tory members would put Brexit ahead of preserving the union.
A YouGov survey of party members found that 63% would be prepared to see Brexit take place even if it meant Scotland leaving the UK.
The SNP said the poll findings were an “utter disaster” for the Scottish Conservative leader.
Ms Davidson said Brexit should be delivered, “but not at the expense of breaking up the UK”.
She told BBC Scotland: “I think there are a number of people within the Conservative Party who need to take a long, hard look at themselves.
“Yes, I understand of course we have got to respect the referendum result, of course we’ve got to deliver Brexit, but not at the expense of breaking up the UK.
“I would remind people of their obligations within the party – yes, we’re a Conservative Party, but we’re also a Unionist party, and I’d remind them that our own union of nations is every bit as important as leaving someone else.”
In other findings, some 26% of Tory members in the poll said they would be happy to see Scotland leave the UK regardless of the outcome of Brexit.
Almost half (46%) said they would be happy to see the Brexit Party’s Nigel Farage as their new leader, while 61% would rather Brexit took place even if it caused “significant damage” to the economy.
SNP depute leader Keith Brown said: “The Tory Party has clearly gone off the deep end – with their Brexit obsession pushing the party further towards the extremes.
“Tory members are so determined to deliver a damaging Brexit they are happy to watch our economy collapse and open the door of number 10 for Nigel Farage to take control of the UK.
“And far from prioritising the union, it’s clear Scotland means so little to the members of the Conservative and Unionist Party that two-thirds are happy for Scotland to become independent if it secures Brexit.
“With this poll suggesting that almost a quarter of Tory members in Scotland would prefer to deliver Brexit for their membership down south even if it means an end to the union, it’s becoming clearer that Ruth Davidson has little authority in her party in Scotland.”
Scottish Liberal Democrat MSP Alex Cole-Hamilton said the poll was a “humiliation” for Ms Davidson.
The survey also recorded that only 39% of Tory members would want Brexit to take place if it meant Labour leader Jeremy Corbyn becoming the next prime minister, with 51% saying they would rather the UK did not leave the EU to avoid that happening.
Scottish Labour deputy leader Lesley Laird said: “The biggest threat to the union today is the Tory Party.
“Their members and their MPs would happily trade tearing our country apart in return for a chaotic no-deal Brexit.”
A spokesman for the Scottish Tories said: “Conservative members need to remember ours is a party of the whole UK.
“It’s in the name. If Scotland becomes independent, there is no UK.
“No matter how badly people want Brexit, no Conservative and Unionist member should be prepared to turn their back on the UK.”