The Colombian midfielder has suffered numerous injuries during a stop-start debut campaign in England, with the latest being a calf knock sustained during the warm-up prior to Everton's 2-1 defeat at home to Aston Villa last weekend.
Manager Carlo Ancelotti insisted in his press conference on Friday that the issue is "minor", though, while revealing that he hopes to have the 29-year-old available for the reverse fixture against Villa on Thursday.
However, there was positive news in the shape of Abdoulaye Doucoure's imminent return from a foot injury.
The former Watford midfielder has not featured since going off injured against West Bromwich Albion in March, and he appears to be in the frame to start against West Ham.
Alex Iwobi, who was Rodriguez's last-minute replacement in the starting lineup against Villa last weekend, could be the man to make way for Doucoure having failed to impress in the defeat.
England defender Michael Keane could also return to the side having missed the previous two matches, with one of Seamus Coleman or Mason Holgate's place in the side potentially under threat should Ancelotti utilise a conventional back four system.
Everton possible starting lineup: Pickford; Coleman, Keane, Godfrey, Digne; Doucoure, Allan, Gomes; Sigurdsson, Richarlison; Calvert-Lewin
Watch the Sports Mole Football Shorts prediction for this game below: