The Red Devils have been heavily linked with the Brazilian over the past few weeks, but have yet to make a formal approach.
Amidst this, Goal claims that the Blues have made contact with his agents, and could compete with United for his services this summer.
The 22-year-old has operated from the right wing throughout his career. He bagged 12 goals and 10 assists from 32 games last term.
He has been touted for a reunion with manager Erik ten Hag at United, but it is unclear whether the club will meet the €70-80m price tag.
United have paid exorbitant fees on players over the past few years, and they seem reluctant to spend over the odds in the transfer market.
This could benefit the Blues, who are likely to go on a £200m spending spree. Antony could be lured with Champions League football.