A procedural hearing held on Friday was adjourned until January 31 to allow "the participants to reach an agreement in principle, if not detail" over the prospect of witnesses giving evidence without retribution.
Separately, the UCI announced its intention to hold a distinct Truth and Reconciliation Commission with the World Anti-Doping Agency to look at doping in professional cycling, after the independent commission investigates.
The independent commission was set up in the wake of the United States Anti-Doping Agency's investigation into Armstrong and the doping practices of his United States Postal Service team, with part of its remit to investigate the relationship between the American and the UCI. Armstrong has since admitted to using performance-enhancing drugs.
The commission, which includes 11-time Paralympic champion wheelchair racer Baroness Tanni Grey-Thompson and is chaired by former Court of Appeal judge Sir Philip Otton, believe an amnesty is required so a witness can provide evidence "without the fear of subsequent disciplinary action against them".
WADA, USADA and pressure group Change Cycling Now agree and will only participate if the amnesty is in place.
The commission also favours an amnesty and negotiations with the UCI, which has accepted it will provide one, will continue until next Thursday.
Should agreement fail to be reached, the inquiry may be delayed.
A statement from the UCI independent commission read: "It appears to the commission that an all-embracing agreement is not a sufficiently real possibility that we should accede to UCI's proposal that we should suspend this inquiry.
"The commission is persuaded that we should allow an opportunity for discussions to continue and for the parties to reach a viable agreement in sufficient detail on an amnesty.
"In the circumstances the commission has decided, with considerable reluctance, that the best course is to adjourn this procedural hearing until Thursday, January 31, 2013, which should be sufficient time for the participants to reach an agreement in principle, if not detail."
The commission expects to be kept informed of the situation by the UCI and next week will decide whether to proceed with its current agenda, timetable and terms of reference.
The statement continued: "There must be no doubt that the commission is concerned to ensure that embarking on an amnesty process does not prevent the criticisms of the UCI in the USADA reasoned decision from being fully investigated with expedition.
"The commission recognises the immense public interest in determining why and how Lance Armstrong and the USPS team were able to engage (as recently, publicly admitted) in systematic doping without detection or sanction."
The UCI proposes a separate inquiry, with an amnesty, with UCI president Pat McQuaid to hold talks with WADA director general David Howman this weekend.
McQuaid said: "We have listened carefully to the views of WADA, USADA and cycling stakeholders and have decided that a truth and reconciliation process is the best way to examine the culture of doping in cycling in the past and to clear the air so that cycling can move forward.
"In addition to contact between our lawyers about establishing a Truth and Reconciliation Commission I spoke to the director-general of WADA earlier this week and I will be speaking to its president over the weekend.
"I welcome the opportunity to work in partnership with WADA on this."