Sky Sports is reporting that Leicester City midfielder Danny Drinkwater is their top target but the Blues have maintained interest in two England internationals.
Arsenal star Oxlade-Chamberlain has entered the final 12 months of his current Gunners contract to prompt talk of a move to Chelsea or Liverpool in the summer transfer window.
The 23-year-old is thought to be weighing up his options at the Emirates Stadium due to concerns about a lack of first-team football at the north London club.
Everton boss Ronald Koeman has already confirmed that Barkley is set to leave the Merseyside club before transfer deadline day later this month.
The talented attacking midfielder has been linked with a potential switch to Premier League runners-up Tottenham Hotspur but his wage demands are proving a stumbling block.
Chelsea finished seven points clear at the top of the Premier League table last season but Antonio Conte is thought to be keen to bolster his squad ahead of a return to the Champions League.
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news