One sport will be added to the programme for the 2020 Games with squash up against karate, the Chinese martial art of wushu, baseball/softball, roller sports, wakeboarding and climbing.
The International Olympic Committee will make its decision on a host for the 2020 Games, as well as which sport to add, at its session in Argentina in September.
Pendleton watched the World Series Squash Finals at The Queen's Club in London with fellow gold medallists Joanna Rowsell and Greg Searle and said the sport had all the ingredients to make it an Olympic favourite.
"Squash is a very physically demanding sport and it's also very spectator-friendly," Pendleton, who won track cycling gold medals in Beijing and London, told the World Squash Federation.
"The nature of the game means that it works really well - it's fast and exciting - and it has all the qualities required to make it a great Olympic sport.
"I don't know why it isn't in already."
Malaysian David beat England's Laura Massaro to retain her title at the weekend, while Egyptian Amr Shabana took the men's title with a win over Englishman Nick Matthew.
Pendleton hailed David's consistency and showed the world number one the keirin gold and sprint silver medals she won in London.
"You so deserve to have one of these," she told the Malaysian.
Rower Searle, who won gold in the coxed pairs at the 1992 Barcelona Games and bronze medals in Atlanta and London, said one of the positives about squash was the game's global reach.
"I think it's really interesting that you can have a sport like squash which is so universal." said Searle. "You have it played in over 150 countries and people of all different shapes and sizes can play it.
"Each sport is different and has different properties. In rowing, if you're not 6 feet 5 it's hard to compete, but squash can be played by everyone."
- Sports & Recreation
- Greg Searle