Adams' manager Nick Cowan added: "Fantastic Guys, awesome and so exciting, so happy for you both, enjoy the moment!!"
A sports journalist posted: "@valerieadams84 Congratulations to you and @GabrielMCPrice Start of a great 2016."
New Zealand Women's Sevens player Charlotte Scanlan messaged: "CONGRATS!! Does that mean no more dirty dancing at the gym? blinging."
A day before the social media post, Price's sister, Josie Elmea Tyquanmo, had posted: "Looking forward to the new addition to my family this year."
That message was initially mistaken for a baby announcement and she posted yesterday: "Oh my goodness!
"Thank you for the baby wishes but my ambiguous status I shared yesterday was about my new sister-in-law Valerie K Adams!"
Adams and Price are yet to confirm when they will tie the knot.
It will be Adams' second marriage.
In August, the shot putter will head to Brazil to chase a third successive Olympic gold medal.