Recent reports in L'Equipe and La Gazzetta dello Sport have suggested Froome faces major competition from Tao Geoghegan Hart, Andrey Amador and Eddie Dunbar for a place in the final eight-rider roster.
Froome has made a full recovery from the terrible injuries he suffered at last year's Critérium du Dauphiné but a lack of race form saw him accept a domestique role at the La Route d'Occitanie and the more recent Tour de L'Ain.
Bernal finished third behind Wout van Aert (Jumbo-Visma) on the uphill stage 1 finish at the Critérium du Dauphiné but Froome eased with 3.8km to go, finishing 5:23 down.
Froome has already confirmed he will move to the Israel Start-Up Nation team in 2021, only fueling rumours he may be left out of the Tour de France squad.
However, Bernal insisted he wanted him in the team.
"Yeah for sure [I want him in the team]," Bernal told British television channel ITV before the start of the opening stage of the Dauphiné.
"He's a rider who has a lot of experience. For sure, to have a seven-time Grand Tour winner in the team is something important. So yes, for sure I would like to have him in the team."
Team Ineos are expected to select their Tour de France squad after seeing how Bernal, Froome and his teammates perform during the mountain five stages at this week's Dauphiné.
Bernal, Geraint Thomas, Michal Kwiatkowski, Pavel Sivakov and Jonathan Castroviejo are expected to be part of the team, with the final three riders confirmed in the countdown to the start of the Tour de France in Nice on Saturday August 29.
Bernal won the La Route d'Occitanie and then battled with Primoz Roglic at the Tour de L'Ain, finishing second overall at 18 seconds. The Dauphiné will be the Colombian's last race before the Tour de France.
"I think the most important goal for us should be to ride as a team and start to feel the real feeling of what we will have at the Tour. We will ride again as a team with a big goal that will be trying to be up there in the GC but looking at the Tour d France more than anything else," Bernal said when asked about his goals for the five-day race, confirming he will target overall victory.
"I think I will try to go for the GC. I think it will be a good opportunity to look in which performance [form] I am in this moment. I will try to do my best and then just prepare for the Tour de France."