With only Paris and Los Angeles left as 2024 bidders after rivals dropped out, IOC President Thomas Bach has raised the prospect of one city being awarded 2024 and the other taking 2028 to avoid there being a loser.
But with less than six months until the vote, the International Olympic Committee is still hesitating over whether to go through with a sudden bidding change.
And Bach's vision to lock up Summer Games hosts for the next 11 years could be derailed by the insistence of both LA and Paris that they are only focused on 2024.
Estanguet, a three-time Olympic champion canoeist who became a member of the International Olympic Committee, is trying to end Paris' run of three losing bids.
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"We're OK if the IOC wants to give two games at the same session in Lima," Estanguet said of the IOC Session in Peru.
"But on our side, our project is only possible for '24. We also believe this is the time to come back to Paris, or to Europe, in '24. This is the fourth bid, we learnt a lot. It's now or never.
LA, which is looking to host the Olympics for a third time, adopted more cautious language. With the IOC yet to confirm the 2028 Games will also be awarded this year, the Americans are only publicly discussing their 2024 campaign.
"Los Angeles is the right city at this critical time for the Olympic Movement and is only bidding for 2024," the LA bid team said in a statement. "With all permanent venues already built and 88 percent public support, only LA 2024 offers the lowest-risk and truly sustainable solution for the future of the Olympic Movement in 2024 and beyond."