International Olympic Committee executives were meeting behind closed doors at their expansive lakeside headquarters in Lausanne, Switzerland to consider the idea.
"I assume it will be a very interesting meeting," said IOC president Thomas Bach.
"We will deal today with some interesting issues, it is such a rich programme."
Bach is a strong proponent of the so-called double vote.
He has complained that there are "too many losers" in Olympic bidding, with cities often spending tens of millions of dollars only to come up empty handed.
While the IOC wants to make bidding less financially punishing, it has also seen the number of countries vying to host Games dry up, as fewer seem willing to take on the vast expense.
IOC leaders are therefore anxious to lock in both Paris and LA – two low-risk, high-quality bids from cities with a record of successfully staging major sporting events.
Bach is due to brief the media following the executive committee meeting.
Friday's expected recommendation would still have to be approved by the IOC's roughly 100 members at a meeting in Lausanne next month, although that body is unlikely to defy guidance from the top brass.
With that hurdle cleared, the IOC's main meeting in Lima on September 13 would then hold a double vote for the 2024 and 2028 Games.
The dual award would all but guarantee a Summer Games for both Paris and LA, as there are no other candidates in the running.
But a fierce battle may continue over which city gets to go first.
Paris last hosted in 1924, and LA in 1984. Both cities have insisted that 2024 is their priority.
The French capital appears to have claimed a lead in the battle for 2024, especially after LA bid chief Casey Wasserman said this week that "LA 2024 has never been only about LA or 2024".
Wasserman said Wednesday then when the idea of double vote emerged, LA shied away from giving the IOC a "now or never" ultimatum.
LA later insisted the bid had conceded nothing in the 2024 race.
Paris has meanwhile held a tougher line.
"We want to get the Games for 2024. That's my mandate", said the co-chief of the Paris bid, Tony Estanguet.
"We can't really consider 2028 at the moment," added the former Olympic champion canoeist.
Also on the IOC's agenda is a change to the bidding process for the 2026 Winter Olympics, with Sion (Switzerland), Calgary (Canada), Innsbruck (Austria) and Stockholm having so far shown interest in hosting.
The IOC will also nail down the scheduling for the 2020 Summer Olympics in Tokyo, where five new sports (climbing, surfing, skateboard, karate, baseball/softball) will be admitted.
The executive committee faces the delicate task of juggling the number of overall events while keeping the number of athletes at 10,500.
In a gender equality drive, there is also expected to be a push for a mixed sex 4x100m relay introduced in athletics and something similar in swimming.
The IOC has also tried to look at ways of reducing the number of events given to sports such as weight-lifting that have been most blighted by doping.