The First Minister wants membership of the European single market and the customs union to be "at the heart" of a new way forward, with the rights of EU nationals living in the United Kingdom guaranteed with immediate effect.
Her party won 35 seats in last week's General Election, down 21 from 2015.
A previous call for a special deal to keep Scotland in the single market was rejected by the UK Government, prompting Ms Sturgeon to demand a second independence referendum when the Brexit process was triggered in March.
Scottish Conservative leader Ruth Davidson, whose party boosted its MPs north of the border from one to 13, has urged the UK Government to listen to those who did not vote Tory and pursue "an open Brexit, not a closed one".
Sturgeon said the United Kingdom government should also reconvene a joint ministerial committee involving all three devolved governments in Edinburgh, Belfast and Cardiff, and set up an advisory groupinvolving Sinn Féin to agree an all-party, all-government negotiating position.
The party also wants a cross-party advisory group to be set up, including representatives from the devolved administrations, to agree a new position for the United Kingdom and oversee the Brexit negotiations.
Scottish Conservatives want emphasis off immigration and on economy
She understands, however, it may not be possible to secure that as part of the Brexit negotiations, the source added. She declined to comment further on Monday.
Ms Sturgeon said a hard Brexit was "no longer viable" after the Conservatives' majority was wiped out in Thursday's vote.
"The Tories' hard Brexit plan has been rejected and we can not allow the Brexit negotiations to become hostage to the inability of either the Tories or Labour to command a clear majority".
She said: "The idea that the United Kingdom led by this Prime Minister and this Government can just blunder into negotiations starting one week today, I just don't think it's a credible proposition".
Scottish Liberal Democrat leader Willie Rennie claimed that an alliance between the DUP and the Tories "risks undermining" the stability of the United Kingdom and urged Mrs May to "swallow her pride" to work with other parties over Brexit.
She said all Scotland's political leaders had endorsed her call for a softer Brexit, which puts heavy emphasis on access to the single market.
"The SNP and the Scottish Government are ready to begin discussions around such a position immediately and I call on all parties to come together to build a new way forward". "And move to a consensus within the country about what it means and what we seek to achieve as we leave".
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