Steven Gerrard is on the look out for a central defender after the departure of Bruno Alves, David Bates and loanee Russell Martin this summer.
The Light Blues have had a £3million bid for Millwall defender Jake Cooper knocked back, while they are also said to be keeping an eye on Blackpool’s Curtis Tilt.
Scouts from Rangers watched Beevers as Bolton took on Girona in a pre-season friendly.
With Bolton’s ongoing financial problems, the 28-year-old would represent a cheap option than Cooper.
Beevers, a former England youth international, has also enjoyed spells at Sheffield Wednesday and Milwall in a career spent playing between the second and third tier of English football.