The former footballer and presenter, 35, has reportedly been signed up to replace Sue Barker, 64, as the host of the long-running quiz show.
Barker is stepping down after 24 years while team captains Matt Dawson and Phil Tufnell will also depart as part of a major shake-up.
Scott posted a photo of a poem from 2018 on her Instagram Stories.
It said: “I was that girl that played football, I really didn’t care / I was that girl that played football, long brown hair / I was that girl that played football, with no regards to the colour of my skin / I was that girl that played football, because I always wanted to win.
“I retired from football with medals and honours galore / I retired from football to open a new door / Now I talk about football live on your TV / Some want to say I’m ticking a box, but that’s not me.
“Football is football and that is what I see. Hi, hello, I’m Alex Scott and you know what, football is me.”
The BBC has denied that any decision has been made over the new presenting line-up for A Question Of Sport.
A spokesman for the broadcaster said: “This is all speculation and no conversations have been had about the future line-up.”
Match Of The Day host Gary Lineker was among those who voiced support for the sportswoman, writing on Twitter on Thursday: “Congratulations and good luck to @AlexScott on being the new host for Question Of Sport.
“Smart, knowledgeable and perfectly qualified for the role. Oh… and if you have a problem with Alex getting the job, you might just be part of the problem.”
On Friday, the tweet appeared to have been deleted.
Barker, Dawson and Tufnell’s final series together will be broadcast next year.
Referred to by the BBC as “the world’s longest running TV sports quiz”, the first edition of A Question Of Sport aired in January 1970.
The move came weeks after the BBC’s director-general Lord Hall was succeeded in the role by Tim Davie, the former chief executive of commercial arm BBC Studios.
Additional reporting by Press Association.