The opposition Social Democratic Party (SDE) on Wednesday proposed that Madise, who has been justice chancellor since 2015, run for president at the election starting in September. Reform's coalition partner, Eesti 200, later also stated it would back a Madise bid.

However, Madise herself has made it clear she has not given her consent to run, a necessary prerequisite to formal nomination.

"We would certainly support her, but first she has to say that she is willing to stand as a candidate," Michal said at Thursday's government press conference, in response to a question from ERR.

"We would be delighted to support her. I personally would gladly back her; I don't think we even need an internal poll. We will make official decisions once there are actual candidates," he added.

Michal echoed comments made by Reform's Riigikogu faction leader Õnne Pillak, who said Madise's name frequently came up in party discussions on potential presidential candidates.

Michal said Madise had also been fair and unbiased in her work as justice chancellor, even as "her decisions and opinions are not always to the government's liking ... But that is the role she has fulfilled honestly."

The prime minister also praised Madise's role in consistently defending Estonia's constitutional values across administrations of varying political hues, which he described as a core presidential duty too.

"She has a thorough understanding of the foundations of Estonia's foreign and security policy, takes part in those discussions and contributes to them," Michal added.

Prime minister: Election process got off to a bad start

Michal said the presidential election process had gotten off on the wrong foot as politicians were hasty to take positions before it was clear whether the incumbent president, Alar Karis, would be seeking another term.

"It is difficult to support someone before there is actually any candidate. The presidential race got off to the wrong start when nobody knew whether the sitting president would run again, yet everyone rushed to take sides and embrace him as if they were already heading into another campaign together. That was a mistake. I hope things go better with Ülle Madise," Michal said.

The prime minister said he expects informal talks between the parties and the sounding out of other potential candidates will take up much of the rest of July, with the Council of Elders likely to meet at the end of the month, or early next month.

"That would then leave about a month to formally nominate candidates and hold the first rounds of discussions," he noted.

While one of Madise's listed plus points is that she could potentially secure a majority in the Riigikogu, eliminating the need for a drawn-out electoral process, Michal said that even if this latter outcome happened it would not represent a crisis.

He noted that in 2016, when Kersti Kaljulaid was ultimately elected head of state, not only did the Riigikogu fail to elect a president from several candidates who initially ran, neither did the regional electoral college – the next phase in the process if parliament draws a blank.

However, the Constitution has all this covered, Michal said. "The mechanism exists, and a president will always be elected."

Isamaa leader: SDE were wrong to name Madise publicly

Urmas Reinsalu, leader of opposition party Isamaa, said that SDE's proposal had forced Madise's hand in stating her position publicly, a move which was "not justified."

At the same time, the justice chancellor is "certainly highly respected" and "is a credible candidate," Reinsalu told "Aktuaalne kaamera."

Center's Riigikogu faction leader found the timing of the SDE announcement, made by its deputy chair Tanel Kiik, "inappropriate."

"We had discussions with different parties, and I got the impression that perhaps we could handle this together without turning it into political spin. But I think this is political spin by the Social Democrats, and it's not fair to the potential candidate either," said Center's Riigikogu group leader, Lauri Laats, echoing earlier comments made by Center Party leader Mihhail Kõlvart.

At the same time, if Madise does elect to run, as a "serious candidate," Center "will certainly begin discussing the matter," Laats added.

In Wednesday's proposal, Kiik did concede that Madise would need to agree to run before she could be nominated.

Madise herself said: "I thank everyone who has expressed support for me, but I have not given my consent to anyone for my candidacy. As Chancellor of Justice, I remain impartial and will continue my work based on that principle."

Estonia's presidents are elected by the Riigikogu in the first instance, and with a two-thirds (68 votes) majority needed in the 101-seat parliament, cross-party support between opposition, coalition and independent MPs is required.

The formal nomination process takes place August 21-24. No other candidate has declared an intention to run, save for the Conservative People's Party of Estonia's (EKRE) co-founder and former leader Mart Helme. Karis announced during the Midsummer holiday he would not be seeking a second term, ending months of speculation on the matter.

The first ballot in the Riigikogu takes place September 2. If after three rounds of voting no head of state has been elected, the process moves to the regional electoral college, which convenes in Tallinn. If this too draws a blank, the Riigikogu's Council of Elders – the speaker, their two deputies, and the heads of the elected parties' factions (currently six) – have the final say.

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