Two individuals whose names have been linked with presidential bids, one by daily Postimees, one by portal Delfi, have denied being contacted on the issue by the coalition parties' leaders.
Former defense minister Jüri Luik is still a possible runner, sources told Postimees at the weekend. However, Luik says he has not been contacted by either Jüri Ratas, Center's leader, or Kaja Kallas, Prime Minister and Reform Party leader.
National Museum (ERM) director Alar Karis, whom Delfi said may be in focus, also denies having been approached by either leader.
Karis told BNS that: "I will not toy with that idea until one of [the two party leaders] approaches me, and then I can say what I think about it."
While Jüri Ratas told Postimees he hadn't met with either individual, Kaja Kallas could not be reached for comment, BNS says.
Two different sources had told Postimees last Friday that Luik, recently appointed Estonia's ambassador to NATO, was to be approached by both Ratas and Kallas, BNS reports. BNS is part of the Postimees Group, along with the daily of the same name.
Ratas, BNS reports, had seen Jüri Luik as a suitable candidate, though Kaja Kallas has told ERR that she was unaware of the recent developments, adding that her party had seen Luik as a viable candidate too.
Kallas said: "We did not have an agreement at this level. It is known that the Reform Party has previously supported Jüri Luik, back when the last presidential elections took place, and in the final round there the Reform Party supported him."
"But this time we did not have such a detailed discussion, because he only just went to NATO as an ambassador, which we decided in the government," Kallas told public broadcaster ERR.
Kallas had said in mid-July that her party had a candidate in mind, but that the individual needed persuading to run.
While current incumbent Kersti Kaljulaid has not ruled out, or in, running for a second term, last week's focus was on academic Tarmo Soomere who, it transpired, will not get sufficient votes from the four Riigikogu parties, Jüri Ratas said last week.
This does not, however, stop Soomere running at the electoral colleges if the Riigikogu draws a blank. Soomere reportedly has one local municipality (Saku) on-side; Ratas has consistently expressed a desire to avoid having a protracted election along the lines of 2016, and says he wants the matter to be settled at the Riigikogu on or soon after August 30.
Ratas himself had been linked in the media with a potential bid, but ruled himself out at the end of May and has more recently reiterated that line.
Of other potential candidates, former justice minister Jüri Raidla gave a flat-out no to Kallas invitation to run, based on a lack of knowledge in some key areas, such as international affairs, while justice chancellor Ülle Madise has not seen any discussion along the lines of party support, though her name was also the subject of media speculation earlier in summer.
The fifth Riigkogu party, the Conservative People's Party of Estonia (EKRE), which is in opposition, has decided to go it alone with its own candidate, former speaker Henn Põlluaas.
Party members have to step down from the party they belong to before taking up the post of president.
In addition to Reform and Center, who do not have enough seats together to reach the 68 or more votes needed to elect a president, the other two opposition parties, the Social Democrats (SDE) and Isamaa, are also very much being sounded out for their stance on any candidate. SDE have tentatively said they would back Kaljulaid for a second term.
Editor: Andrew Whyte