The two most senior posts in the Scottish judiciary are yet to be formally awarded but the two judges are understood to be the current favourites.
The news follows the revelation in The Herald yesterday that Colin Boyd, now Lord Boyd, and Maggie Scott, QC, are expected to be appointed high court judges.
Lord Hamilton, the current Lord President and Scotland's most senior judge, is due to retire in June. A formal announcement about his replacement is not expected until next month.
The First Minister established a selection panel in January to make recommendations for a replacement. The panel includes Sir Muir Russell, chairman of the Judicial Appointments Board for Scotland.
The Lord President is the most senior Scots judicial position and the office holder is responsible for leadership of the Scottish judiciary, in addition to chairing the board of the Scottish Court Service. The salary is £214,165.
Lord Gill was made a judge in 1994. He was appointed Lord Justice Clerk and President of the Second Division of the Inner House in November 2001.
He is perhaps the most obvious contender. However, he could still be passed over because of his age. He is already 70.
Lord Carloway recently chaired a review of the Scottish criminal justice system commissioned by the Scottish Government. It made scores of recommendations, such as calling for the end of corroboration. He was made a judge in 2000.
In 2004, he presided over the prosecution of gas transporter Transco under health and safety legislation for an explosion in Larkhall, South Lanarkshire, which killed a family of four, fining the company £15 million.
A Scottish Government spokesman said: "The recruitment process for the appointment of the new Lord President began earlier this year and a decision will be announced in due course."
Meanwhile, the news Lord Boyd is due to become a judge has provoked criticism.
He led the prosecution of the man convicted of the Lockerbie bombing, and was criticised by the Scottish Criminal Cases Review Commission for failing to disclose information to Abdelbaset Ali Mohmed al Megrahi's defence. He rejected the claims.
When he resigned as Lord Advocate in 2006 there was speculation his decision was connected to the Shirley McKie fingerprint inquiry and Lockerbie case criticisms. He denied that and said it was just time to move on.
Ms McKie's father Iain said the news "took my breath away". He said: "That the Judicial Appointments Board should recommend to the First Minister that Lord Colin Boyd be appointed as a judge to the highest court in the land set new standards for being out of touch."
Tam Dalyell, the former MP and father of the house, said: "The fact he may well become a judge should not inoculate Lord Boyd from the obligation to answer questions on Lockerbie over the period that he headed the Crown Office.
"The Crown Office, and I would have thought Lord Boyd in his position in the Crown Office, have an obligation to address powerful criticisms of non-disclosure."
Of the others expected to become judges, Ms Scott, who was sacked by Tommy Sheridan during his perjury trial in 2010, led Megrahi's recent appeal case and the defence in many high-profile cases, including that of Luke Mitchell, who was convicted of murdering teenager Jodi Jones in January 2005.
It is thought that Michael Jones and David Burns have also been recommended for the position of judge.