The Eagles boss handed Ings his one and only England cap when he was in charge of the national side and is known to be chasing a regular goalscorer to increase the potency of his side up front.
Palace are expected to face stiff competition for the signature of the 25-year-old, with Newcastle United, Burnley and West Ham also credited with an interest, according to reports on Wednesday.
Palace, West Ham and Newcastle were all linked to Ings in December, shortly after the striker had just made his first appearance for Liverpool in 11 months after recovering from a knee injury.
Anfield boss Jurgen Klopp refused to let the ex-Burnley striker leave on loan in January, but the German is understood to be more sympathetic to Ings' desire to play regular first-team football again next season, which he seems unlikely to do for the Champions League finalists.
Palace played without a recognised striker for an uncomfortable period last season when Christian Benteke was out through injury and Hodgson will be eager to go into next season with better options up front.