The 28-year-old has attracted interest from a host of Premier League clubs following West Brom's relegation from the Premier League.
According to The Sun, Livermore had looked set to sign for Steve Bruce at Aston Villa, however any move was dependant on the Birmingham-based side securing promotion to the Premier League.
With Aston Villa missing out on a Premier League return at the hands of Fulham in last week's Championship play-off final, Hodgson now has a clear run at the England international.
The Palace boss handed Livermore his international debut six years ago, however it wasn't until Gareth Southgate's reign that he has featured regularly for the Three Lions.
Livermore is a graduate of the Tottenham academy but spent the majority of his time out on-loan across the country, appearing for MK Dons, Crewe Alexandra, Derby County, Peterborough United, Ipswich Town, Leeds United and Hull City.
The central midfielder eventually made a permanent move away from the north London club in 2014, signing for Hull City - where he had spent the previous season on loan.
Following three years and 144 appearances for the Tigers, Livermore became a £10m signing for West Brom in January 2017.
Livermore is currently with the England squad preparing for the World Cup. The midfielder is on the standby list and will travel with the squad to Russia should a member of the Three Lions suffer an injury.