The band, whose cover of The Beatles’ Golden Slumbers is the soundtrack to this year’s clip, have been given odds of 6/1 by Paddy Power to reach the summit on the chart issued December 22.
Only George Michael (13/8), Ed Sheeran (4/1) and Mariah Carey (4/1) have better odds.
William Hill have also narrowed their odds down to 6/1 from 100/1, ahead of this year’s X Factor winner (9/1).
A spokesperson said: “Elbow are not associated with Christmas and as such were massive outsiders.
“We think the song is very good and a genuine contender for the festive number one.”
The song has proven to be a very popular element of the new commercial.
“Love the @johnlewisretail ad but I love this version of #goldenSlumbers by @Elbow even more. @Guy_Garvey ‘s voice never fails! #mozTheMonster,” one person tweeted.
Another wrote: “Not as much of a tear jerker as the old man, penguin, or bear but still good and obvs love the soundtrack by @Elbow (Guy Garvey should just be the voice of Christmas full stop)! #MozTheMonster.”
A third said: “#MozTheMonster Elbow can do no wrong and their version of Golden slumbers is perfect, the advert is fun and we don’t want to be balling our eyes out every year, this one is touching and clever.”
If the Moz the Monster-endorsed Golden Slumbers does go all the way, it will be the first John Lewis Christmas ad song since 2013 to do so.
Lily Allen’s Somewhere Only We Know was the last song on the retailer’s festive treat to hit No1, with last year’s track – One Day I’ll Fly Away by Vaults – going no higher than No53.
Gabrielle Aplin’s spin on The Power of Love (2012) is the only other track from the ads to hit the top spot.
Ellie Goulding’s Your Song hit No2 in 2010, while Slow Moving Millie’s Please, Please, Please, Let Me Get What I Want charted at No31.
Tom Odell’s Real Love peaked at No7, while in 2015 Auora’s Half The World Away was No15.
The John Lewis advert has just been released, meaning Elbow’s effort is expected to scale the iTunes rankings over the coming hours.
Currently the digital store’s rundown is dominated by Rita Ora’s Anywhere (No1), Camila Cabello’s Havana (No2) and Ed Sheeran’s Perfect (No3).
Last year’s Christmas Number One was Clean Bandit’s Rockabye.
It was reported yesterday that Michelle McManus, former Pop Idol winner, is also in this year’s race with a charity single.
Elbow have never had a number one single, and have in fact only graced the Top 10 once: with 2008’s One Day Like This, which reached No4 several years after its release thanks to use at the London Olympics.
They’ve had much more success with their albums: their last two records, 2014’s The Take Off And Landing Of Everything and this year’s Little Fictions both hit the summit.