After talks in Downing Street, she said she was not prepared to see "Scotland driven off a hard Brexit cliff- edge".
She and the first ministers of Wales and Northern Ireland have demanded to be fully consulted on Brexit negotiations with the EU.
But Downing Street said it was vital not to "undermine" the UK's position.
The Scottish and Welsh leaders want the devolved legislatures and Parliament, all to vote on the UK's approach when formal talks begins with the European Union about Brexit.
The issue was discussed as the joint ministerial committee - bringing togetherMay, Sturgeon, Welsh First Minister Carwyn Jones and Northern Ireland's First Minister Arlene Foster and Deputy First Minister Martin McGuinness - met for the first time since 2014.
Asked about calls for different parts of the UK to be able to opt in or out of the single market, the prime minister's official spokeswoman said a united UK negotiating position was "vital to protect the UK's interest as a whole".
She said : "I'm not going to stand by and watch Scotland driven off a hard Brexit cliff-edge. Scotland's vote to stay has to be respected.
"There's not a bit of what I'm doing just now that's bluffing or game-playing. This is not a game of chicken. It's not a game at all."
She told reporters: "If all we get from the negotiations is a door slammed in our face, Scotland should be able to chose a better option."
Foster, also speaking in Downing Street after the meeting, said: "It's important that we are involved in the heart of [the Brexit negotiation] process.
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