The alleged agreement will result in the 34-year-old appearing on the BBC every day once Strictly returns on September 11.
"She's really enjoying her time at the BBC and has taken to The One Show like a duck to water," a source told The Mirror.
"Alex loves getting stuck in and being offered Strictly is another chance to show how game for a laugh she is. It's a dream come true and shows how the BBC has great faith in her."
The report comes less than a week after the BBC was said to be seeking a replacement for Jones on The One Show, which has suffered a drop in ratings since her appointment as host last August.
However, the corporation told Digital Spy that claims of concern about Jones's perceived cold persona were "categorically not true".
"Alex is exactly what the BBC should be about, developing young talent and turning them into stars," a BBC spokesperson said today.
Jones's One Show co-host Matt Baker finished second to EastEnders actress Kara Tointon in last year's Strictly Come Dancing.
Jones has been rumoured to be appearing on the forthcoming ninth series of the show since February.