The 38-year-old singer - who has 20-month-old son Ever with husband Mario 'MC Souleye' Treadway - is said to be "excited" at the prospect of joining the show, especially because it would help boost sales of her forthcoming album 'Havoc and Bright Lights', which is released later this month.
A source told RadarOnline.com: "Alanis is really excited about the prospect of being on the show.
"She has a new album coming out and the publicity would be great. She kind of disappeared for a while to get married and have a baby, but now she's ready to get her career back on track."
Alanis is seen by show bosses - who recently unveiled Mariah Carey as a new panellist - as a perfect replacement for departed judge Steven Tyler, who quit the talent contest along with Jennifer Lopez last month.
The source added: "They have Mariah Carey for the diva figure on the show and Alanis would be the honest judge who could really teach the contestants important lessons about the music business.
"Nothing is signed yet, but they're negotiating and near closing a deal."
If Alanis gets the nod to sit alongside Mariah and Randy Jackson, Nick Jonas could be in for a disappointment as he recently confirmed he is in talks for a role on the show.
He tweeted: "The rumors are true ... I am being considered to be a judge on American Idol, and it would be a dream come true if it happens. #nickonidol (sic)."
However, Randy - the only remaining original judge - is rumoured to be moving to a mentoring role on the show, which would leave another seat vacant.