The Blues are keen to strengthen with a defensive midfielder if the transfer window goes ahead this summer and are admirers of the England international, whom they released as a youth.
Mount was recently forced to apologise for playing football with Rice at a north London centre while he was supposed to be self-isolating, after team-mate Callum Hudson Odoi was diagnosed with coronavirus.
Former Republic of Ireland youth international Rice has been a consistent figure for West Ham and bagged two assists in 29 Premier League appearances before football was suspended because of the coronavirus pandemic.
He would not come cheap, with four years left on his West Ham contract and the option of a further year's extension.
It's remains uncertain whether the UK transfer window will open as scheduled on June 18, with no definitive date set for when the current domestic season will be resumed amid a global health crisis.