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In a letter to the prime minister they said they would not support any deal that introduces different arrangements for Northern Ireland.
They said this would "undermine the integrity" of the UK.
Scotland's Brexit Minister Mike Russell said it was "an astonishing development".
Sources have told the BBC Ms Davidson and Mr Mundell have made it clear they would resign if Northern Ireland faces new controls that separate it from the rest of the UK - because that could fuel the case for Scottish independence.
A source close to Mr Mundell told the BBC: "If you find yourself not agreeing with government policy" resigning would be the "logical outcome". A source close to Ms Davidson said the issue was a "red line".
In the letter to Theresa May, the Scottish Tory leader and Scottish Secretary said: "Having fought just four years ago to keep our country together, the integrity of our United Kingdom remains the single most important issue for us in these negotiations. [...]