Francis Maude also claimed that a breakaway Scottish party would have a better chance of defeating the SNP’s planned independence referendum.
Mr Maude, a Cabinet Office minister and key ally of David Cameron, has previously indicated he could support such an idea.
The proposal, by one of the frontrunners to become the next party leader in Scotland, has provoked a furious row within the Scottish Tories.
Party insiders admit at the last General Election the more money they poured into some Scottish constituencies the more the electorate were motivated to vote for other parties.
Despite being in power in Westminster, in coalition with the Liberal Democrats, the party has just one MP in Scotland.
Mr Fraser’s plans were attacked by Sir Jack Harvie, the transport tycoon, who has been the party’s largest backer in Scotland.
However, they found backing from another multimillionaire last weekend.
Alister Jack said that “significant” sums would be available to the new grouping if Mr Fraser wins the race to take over from Annabel Goldie.
Mr Maude appeared not only to endorse the idea yesterday but to suggest it would help to defeat the SNP’s independence poll.
Speaking to journalists at Westminster, he said: “You could make the case that when it comes to a referendum on independence, the case for Scotland remaining part of the United Kingdom is more effectively made by the party which is incontrovertibly and unambiguously Scottish, rather than one that still has in Scotland a bit of baggage of looking like the tail end of an English party.”
He added the Conservatives in Scotland needed to work out the answer to their electoral problems, not have a solution suggested from London.