Sources said that Justices Gogoi and Lokur signed the joint two-line letter on April 22 in which they spoke about holding of a "full court".
The same issue was for the first time raised by Justice J Chelameswar on March 21, which was followed by a similar missive on April 9 by Justice Kurian Joseph who had sought the setting up of a bench of seven senior most judges to deal with the issues plaguing the top court.
The short letter by Justices Gogoi and Lokur urged the CJI to convene a full court on the judicial side to discuss institutional issues and the "future" of the top court, they said.
As per convention, a full court meeting of the Supreme Court, involving all judges, is usually convened by the CJI when a matter of public importance relating to the judiciary comes up.
The Monday morning tea meeting of all the judges was held just after Naidu had announced rejection of the impeachment notice. The sources said the CJI did not say anything about the outcome of the meeting, especially relating to the full court.ALSO READ: Naidu rejects CJI impeachment notice; Cong to move SC challenging decision
However, Justices Gogoi and Lokur were of the view of leaving behind the issue of impeachment and moving forward by holding a discussion among the judges to overcome the issues plaguing the highest judiciary. Justice Gogoi is the next in line to take over the mantle of the CJI from Chief Justice Misra who retires on October 2.
Irked over the government's delay in clearing the Collegium's recommendation to elevate a judge and a senior woman advocate to the apex court, Justice Joseph had also written to the CJI contending that the "very life and existence" of the institution was "under threat" and a "surgical intervention" is required.
He had requested the CJI to set up a bench of seven senior-most judges to take up the matter of appointments to its logical conclusion.
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In his letter to all the judges on March 21, Justice Chelameswar had urged the CJI to convene a full court to take up the issue of alleged executive interference in judiciary.