Swinney and Forbes considering bids for SNP leadership

He is currently the favourite to replace Humza Yousaf.

Leading bookmaker Coral has already suspended betting on who will be the next first minister "following strong sustained support" for Mr Swinney.

READ MORE: Inside story: Humza Yousaf's resignation 

One supporter of the Perthshire North MSP described him as the “safest pair of hands in Scottish politics.”

However, one MP said they feared he would take the party "backwards."

In his resignation speech, Mr Yousaf said he had told Lorna Finn, the SNP's national secretary, of his intention to stand down as party leader and asked that she "commences a leadership contest for my replacement as soon as possible."

It is understood the party's ruling NEC will meet later this week to set out a timetable for the vote.

Mr Yousaf said that he would remain as First Minister until his successor had been elected.

Mr Swinney was in London, appearing at an event marking 25 years of devolution held by the Resolution Foundation.

Asked by broadcasters if he would run for leadership, he said he would "give it consideration."

"I've got lots of things to think about. There's the whole question of my family and I have to make sure that I'm doing the right by my family, they're precious to me.

"I have to do the right thing by my party and by my country.

"So there's lots to be thought about. And I'll give all of that consideration in the days to come."

Mr Swinney appeared to offer an olive branch to the Scottish Greens. He said there had clearly been "strains within the Bute House Agreement, and it's important that these issues are considered carefully and handled with care."

"And obviously, I spent a lot of my time in government doing exactly that in the past," he added. “So it's important that those are the values and the characteristics that are brought to how we deal with other political parties, particularly now that the SNP will be a minority government in the Scottish Parliament.”

There is broad support for Mr Swinney amongst the Holyrood group, who are keen to avoid a repeat of last year’s deeply divisive leadership contest.

Three other potential candidates Neil Gray, Jenny Gilruth and Westminster leader Stephen Flynn have all ruled themselves out, all backing Mr Swinney.

Mr Flynn described him as a “serious man for serious times.”

“I think the opposition parties in Holyrood would be very worried about someone with his particular abilities taking on the role, because he will be able to show a level of leadership which is unmatched in Holyrood,” he added.

READ MORE: John Curtice: The two things the next SNP leader will need to do

Mr Swinney served as the SNP's leader between 2000 and 2004 before being ousted by unhappy backbenchers after a string of disappointing election results.

Sources close to Kate Forbes have suggested she too is thinking about it, though they say it is very early days.

“I guess if Swinney decides to stand maybe nobody else will put their hat in the ring, in which case it might be a coronation, which is certainly better than a divisive leadership contest,” Prof Sir John Curtice told The Herald.

“But then Swinney has to steady what is still a difficult ship because there's still the police allegations hanging over [the SNP]. And he’s very much associated with the Sturgeon regime, etc, etc.

“But at least he's got the experience. At least he might make a reasonable fist of it, but they may still find themselves struggling.”

However, SNP MP Joanna Cherry said that Mr Swinney would not take the party in the right direction.

She tweeted: "John Swinney is hugely respected across our party but the lesson of the last year is that the SNP needs a complete reset. We must go forward not backwards.

"Kate was right when she said that continuity would not cut it. The next leader must deliver change."