The former footballer and pundit, 35, played down suggestions she had already signed up to replace Sue Barker, 64, on the show.
The BBC has denied that any decision has been made over the new presenting team, saying “no conversations have been had about the future line-up”.
After Scott was linked to the role, she was inundated with comments that it was a box-ticking exercise. Others said it was “her colour that gets her the job”.
She wrote on Twitter: “The love and support shown to me yesterday has me smiling through the weekend, thank you.
“I am going to #keeponkeepingon like always!
“The only thing that is signed sealed delivered that I am buzzing about right now though is Pierre-Emerick Aubameyang at Arsenal,” she said, referring to the Arsenal captain signing a new deal on Tuesday.
Scott also shared an anti-racism poem from 2018 on her Instagram Stories, with the line: “Some want to say I’m ticking a box, but that’s not me.”
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.
On Thursday, Lineker wrote: “Congratulations and good luck to Alex Scott 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.