Asher-Smith swept to a three-medal haul at the IAAF World Championships in Doha last summer, including the 200m title, fuelling hopes she could stage a repeat at the Olympics.
However, Thompson says the Olympics are "five levels above" the World Championships and said just one medal would be a success – especially now the Games have been pushed back a year.
"It's so difficult to go out there and win just one Olympic gold medal, even a silver or bronze. The Olympic Games are so different from a World Championships," Thompson told Eurosport.
"In the Olympics, people are prepared to die. It’s the most important thing. The amount of effort athletes put into it is huge. It’s a completely different magnitude.
"Dina has acquitted herself well, but the Olympics is the next step. And she's still got to take that step and perform.
"One thing we love about sport is that nobody gives it to you. It doesn't matter that you're the world champion or the world's fastest, you've got to prove it again on the day."
Dina Asher-Smith of Great Britain celebrates after winning the Women's 200m Final during day six of the 17th IAAF World Athletics Championships Doha 2019 at the Khalifa International Stadium in Doha, Qatar. (Photo By Sam Bar
Image credit: Getty Images
Asher-Smith has said she may not compete in 2020 rather than "racing for racing's sake" and risk catching coronavirus.
The 24-year-old is expected to compete in the 100m, 200m and 4x100m relay in Tokyo.
And Thompson, speaking to promote his podcast series on the Olympic Channel as Bridgestone Olympic ambassador, said the unpredictability of sprinting made Asher-Smith's challenge even trickier.
"Dina had a brilliant year last year. Of course she’s going to be among the favourites [in Tokyo], but there are some other really good girls out there," Thompson said.
"One of the things about the sprints is that there could be a youngster, say aged 16, somewhere in the world who has unbelievable ability. And now they're going to be another year better and are just going to pop up.
"It’s quite possible that a winner or medallist in the 100m or 200m is going to be someone we've never heard of. That can happen in the sprints."
Katarina Johnson-Thompson clinched gold at the World Championships
Image credit: Getty Images
Thompson also backed Katarina Johnson-Thompson to build on her heptathlon world title next summer – even if he conceded she would probably not be favourite for gold in Tokyo.
"I think KJT’s prospects in Tokyo are really good after her World Championship win. I think she now has the confidence to know that providing she performs well, she’s in with a chance," Thompson added.
"But the thing about the heptathlon is the best athlete in the world is Belgium’s Nafi Thiam. If both of them are at 100%, I would say Thiam is a slight favourite.
"But having said that, if the Olympics awarded medals to people just because they're favourites, then no one would bother turning up. You have to earn it. So if KJT performs well, she can put Thiam under pressure and who knows how she’ll respond."
Johnson-Thompson comfortably beat 2016 Olympic champion Thiam in Doha, setting a British record of 6,981 points in the process.
Daley Thompson is an ambassador for Worldwide Olympic & Paralympic partner, Bridgestone for their ‘Chase Your Dream, No Matter What’ Olympic campaign. His new podcast series on the Olympic Channel is out now! To listen, click here.