After a record result in the Scottish local government elections, where they returned 276 councillors, the Conservatives now have seats such as Mr Salmond’s Gordon constituency and the Moray seat of SNP depute leader Angus Robertson in their sights.
Ms Davidson, the Scottish Tory leader, has already declared her party has a “pretty good” chance of winning the Moray constituency.
And on Saturday she campaigned in Gordon, where the former first minister and SNP leader is the candidate.
Ms Davidson insisted then: “We won the local government election is Gordon this week, beating the SNP into second place. It means that in this seat, as in many others, it is a two-horse race between us and the Nationalists.
Mr Salmond conceded that support for the Conservatives is on the rise in Scotland - but claimed “presumptuous announcements” by the Tories would only seek to boost support for the SNP.
“There’s no question the Tory vote is on the increase and therefore we have to galvanise the SNP support,” he told Press Association Scotland.
“The Tories are helping us in their rather presumptuous announcements that they’re going to win the seat.
“I have to say I’m delighted, the effect is to galvanise the SNP support.”
Speaking in Ellon as he launched his campaign to be re-elected on June 8, Mr Salmond stated: “I said the Tory support was on the rise, it should be said in Gordon constituency we got more councillors in than the Tories, we got 11 councillors to their nine and six for the Liberal Democrats.”
The former first minster continued: “I’ve never taken any election battle for granted, I’ve represented the north-east of Scotland for 30 years and every election the Tories tell me they are going to beat me and every time they fall short.
“Also in the north-east of Scotland it’s not the greatest thing to boast you’re going to win beforehand, the folk here tend to bring them back down to earth with a bump
“And that I think will happen to the Tories. Boasting before an election is an extremely foolish thing to do. The people have their say at the ballot box and I think the north-east of Scotland has a way of bringing people who make vainglorious boasts down to earth with a sharp bump.”