The presenter, 42, is said to be the "first choice" to replace the TV veteran after he officially quit the show on Friday.
"The job is Claudia's for the taking," a source told the Daily Star. "If she turns it down again the only other favourite would be Alan."
The insider revealed that Tess had a different replacement in mind – her husband Vernon Kay.
"For Tess, having Vern by her side would've been brilliant. She'd have been able to enjoy a similar on-screen relationship with him as she did with Bruce," they added.
"It's a lot of pressure to be the main presenter for the live show and they'd prefer Claudia to take up that role which could leave Tess feeling a little put out. She's quite worried about what Bruce's departure means for her."
Do you think Claudia would be a good replacement for Bruce? Scroll down to place your vote below.
On Friday, Bruce confirmed his departure from the show after 11 series with a series of tweets on his Twitter account.
"After 10 wonderful years & 11 series, Sir Bruce Forsyth has decided to step down from the rigours of presenting our live shows", they tweeted, before adding: "Sir Bruce is one of the great showbiz legends of our time and our success is due in vast amounts to him. This is not farewell!"
The presenter and entertainer, whose career has spanned 74 years, will continue to present the special episodes of the show, including the Christmas and Children In Need specials.
In a statement released by the BBC, Sir Bruce said: "I am very proud of what the show has achieved and confident it will entertain the nation for many years to come. I am also delighted that by presenting the Christmas and Children in Need shows I will continue to have a strong association with Strictly."
BBC One controller Charlotte Moore called Sir Bruce a legend and credited the 86-year-old with the show's success.
"Sir Bruce Forsyth is one of the great showbiz legends of our time and Strictly's success is due in vast amounts to him", she said. "I am so pleased he will continue to be part of the Strictly family and promise viewers that we haven't seen the last of him on BBC One."
The British television show features professional dancers partnered with celebrities from around the world competing in ballroom and Latin dances.
Primarily running on Saturday evenings, Sir Bruce has been joined by Tess Daly in presenting the show.
There has been no announcement about who will replace him on the show however odds are already on who will be Tess's new co-host.
Anton du Beke is the favourite at 7/4, followed by Graham Norton, John Barrowman and Claudia Winkleman.
- View results
Yes, she is perfect!
No, a different choice would be better
- Return To Poll