The party wants to see more money for the NHS, police and higher education, as well as a commitment not to further increase taxes on working Scots.
Tory MSPs said that would be the price for Scottish Conservative support for Mackay, who will set out his draft budget for the year ahead on February 6.
The SNP relied upon Green votes to pass its last set of spending plans in 2019. But the cost of that deal was a Scottish Government commitment to hand powers to councils to introduce a workplace parking levy or a new tourist tax on hotel stays, if they so wished.
Tory MSP Murdo Fraser has also called for all budget increases arising from increased NHS spending in England to be passed onto the health service here.
That would fund more drug rehabilitation beds, refunds for NHS staff who have to pay at hospital car parks, and the redressing of the controversial resource allocation system.
The plans would also mean a £50 million injection for police, no cuts to council budgets and a reduction in business rates.
Fraser said: “Although the UK budget will not be published until March 11, there was never anything to prevent the Scottish Government going ahead with producing its own budget in advance.
“We welcome the timetable put forward by the finance secretary for the budget process, which puts into perspective the faux outrage that he expressed at the timing of the UK budget.
“There can be no justification for additional tax rises, or further cuts to public spending, against this backdrop.
“The two areas that we view as priorities for this budget are measures to grow the Scottish economy, and support for vital public services.
“We will assess any budget proposals from the SNP government against these priorities.”
The SNP has relied upon Conservative support to pass its Holyrood budget in the past. In its 2007 minority government, the party made several deals with the then Tory leader Annabel Goldie.
Responding to the Tory budget offer, Scottish Greens co-leader Patrick Harvie said: “It’s no surprise that the Tories want to ignore the climate emergency while lowering taxes for their friends, but this offer does present Derek Mackay and the SNP with an interesting choice.
“He can work with the Tories, as the SNP did in their first term in government, or he can stay on the progressive path the Greens have introduced, resist the toxic agenda imposed by the UK government and work constructively with us to tackle the climate emergency head-on and build a new progressive Scotland.”