The Red Devils have expressed interest in signing Sterling, who has been told to leave Stamford Bridge by new Blues boss Enzo Maresca.
According to Sky Sports, Manchester United are also exploring a deadline-day move for Chelsea midfielder Carney Chukwuemeka, another player who the west Londoners could sell before the 11pm deadline.
Chelsea signed Chukwuemeka from Aston Villa in the summer of 2022 and he played 14 Premier League games in his first season at Stamford Bridge.
The 20-year-old found first-team opportunities harder to come by last term, however, and has now featured in any of Chelsea's matchday squads under new manager Maresca.
Chelsea had been in talks to sell Sterling to Manchester United but the winger's most likely destination now appears to be Arsenal.
The Blues are still expected to agree a deal with United, however, with Jadon Sancho edging closer to a surprise move to Stamford Bridge.
Chukwuemeka is one of several players Chelsea are willing to offload before the summer transfer window closes and Jamie Redknapp admits he 'feels sorry' for several within the Stamford Bridge squad.
'There have been things that I obviously don't like, with things that have obviously happened on the matchday - when Sterling put out the statement, I thought that was inappropriate.
'When you've got so many players you're not going to keep everyone happy and that's the predicament they have.
'You cannot force a player to leave. Some players will want to play some will be happy to sit on the bench and just train. it doesn't help the atmosphere at the club.'
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