Sources said the opposition BJP is adamant that the Lokayukta should be removed though it may not press on recall of Governor Kamla Beniwal.
Mukherjee, who is Leader of the House in Lok Sabha, and Advani held a meeting in Parliament House over the issue after Parliament was adjourned for the day following disruptions by BJP over the issue.
"Advani suggested certain things about the Gujarat Lokayukta," Mukherjee said after the meeting but did not elaborate.
Soon after this meeting, Advani held a meeting with his colleagues, including Leader of Opposition in Lok Sabha Sushma Swaraj and her Rajya Sabha counterpart Arun Jaitley.
One option being explored by the two sides is to make Lokayukta R A Mehta put in his papers on some pretext, sources said.
Prime Minister Manmohan Singh and Mukherjee have agreed to the removal of Mehta and have even conveyed this to the BJP top brass during the last few days, BJP sources claimed.
The main opposition is now saying it is time that the government kept its word and removed Mehta if it wants to end the deadlock.