Livermore had been expected to leave the Baggies following their relegation from the Premier League last season, but the England international played the full 90 minutes of both of his side's opening two Championship outings.
Sky Sports News reports that the 28-year-old could yet be offered a late route out of The Hawthorns by Watford, who are hopeful of pushing through a deal before today's 5pm transfer deadline.
The Hornets are understood to be lining up an initial season-long loan deal for the midfielder, with the option to make that move permanent next summer.
Watford were originally eager to bring Southampton midfielder James Ward-Prowse to Vicarage Road, but have now switched targets after being rebuffed by the Saints.
Livermore joined West Brom in January 2017 and has since gone on to make 54 appearances for the club.