Just 10% of Brits said they wouldn't support their child if they changed sex, with 69% of people stating they would support their child and 21% said they weren't sure how they would feel.
The exclusive YouGov research, which polled 1,672 people, also found a quarter of Brits would be happy if their child wanted to change sex.
The poll was part of HuffPost UK's TransBritain series.
The results revealed that age played a role in people's opinions of transgender children, as 47% of those aged 18 to 24 said they would be happy with their children changing sex, but only 17% of those aged 60 and over said they would be happy with a child transitioning.
Looking at regions, the poll revealed Londoners would be the least approving, with 41% saying they'd be unhappy with the outcome.
Overall, 36% of people polled said they would be unhappy with their child changing sex and 39% were "not sure".
Joan King from The Gender Trust welcomed the survey results.
King told HuffPost UK Parents: "From my point of view I am not surprised by your findings because there is a great deal of information available on the internet and from organisations like ourselves.
"The subject has also been in the storylines of a number of TV series (Hollyoaks, A touch of Frost, and Coronation Street) plus a number of documentaries on the subject.
"The subject is more openly discussed today and there are more individuals seeking help and support."
Siobhan Freegard, founder of video parenting site Channel Mum, is a mother who would support her child if she changed sex.
She told HuffPost UK Parents: "You love your child whatever their hair colour, their eye colour or their character, so why should their gender make any difference?
"It’s true that some parents may struggle with the idea of their child changing sex, but this may be more to do with mums and dads re-adjusting their hopes and dreams for their child from their birth gender, to the person their child chooses to be."
Freegard said she understands changing sex is a huge step but it is also a positive one.
"If your child chooses this route, it’s a testament to your parenting that they are strong enough and confident enough to make this decision," she added.
"Celebrate who your child is and give them the love and support they need."
Useful websites and helplines:
- The Gender Trust supports anyone affected by gender identity | 01527 894 838
- Mermaids offers information, support, friendship and shared experiences for young people with gender identity issues | 0208 1234819
- LGBT Youth Scotland is the largest youth and community-based organisation for LGBT people in Scotland. Text 07786 202 370
- Gires provides information for trans people, their families and professionals who care for them | 01372 801554
- Depend provides support, advice and information for anyone who knows, or is related to, a transsexual person in the UK